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Lessons In Industrial Instrumentation 

By Tony R. Kuphaldt 
Version 0.2 - Released September 29, 2008 

© 2008, Tony R. Kuphaldt 

This book is licensed under the Creative Commons Attribution License, version 3.0. To view a 
copy of this license, turn to page 631. The terms and conditions of this license allow for free copying, 
distribution, and/or modification of all licensed works by the general public. 

Revision history 1 

• Version 0.1 - July to September 2008 (initial development) 

• Version 0.2 - released September 29, 2008 for Fall quarter student use 

1 Version numbers ending in odd digits are developmental (e.g. 0.7, 1.23, 4.5), with only the latest revision made 
accessible to the public. Version numbers ending in even digits (e.g. 0.6, 1.0, 2.14) are considered "public-release" 
and will be archived. Version numbers beginning with zero (e.g. 0.1, 0.2, etc.) represent incomplete editions lacking 
major chapters or topic coverage. 



Physics 7 

1.1 Terms and Definitions 8 

1.2 Metric prefixes 9 

1.3 Unit conversions and physical constants 10 

1.3.1 Conversion formulae for temperature 13 

1.3.2 Conversion factors for distance 13 

1.3.3 Conversion factors for volume 13 

1.3.4 Conversion factors for velocity 13 

1.3.5 Conversion factors for mass 13 

1.3.6 Conversion factors for force 13 

1.3.7 Conversion factors for area 13 

1.3.8 Conversion factors for pressure (either all gauge or all absolute) 13 

1.3.9 Conversion factors for pressure (absolute pressure units only) 14 

1.3.10 Conversion factors for energy or work 14 

1.3.11 Conversion factors for power 14 

1.3.12 Terrestrial constants 14 

1.3.13 Properties of water 15 

1.3.14 Properties of dry air at sea level 15 

1.3.15 Miscellaneous physical constants 15 

1.3.16 Weight densities of common materials 16 

1.4 Dimensional analysis 18 

1.5 The International System of Units 19 

1.6 Conservation Laws 20 

1.7 Classical mechanics 20 

1.7.1 Newton's Laws of Motion 21 

1.7.2 Work and Energy 22 

1.7.3 Mechanical springs 25 

1.8 Fluid mechanics 27 

1.8.1 Pressure 28 

1.8.2 Pascal's Principle and hydrostatic pressure 33 

1.8.3 Fluid density expressions 38 

1.8.4 Manometers 40 


1.8.5 Systems of pressure measurement 43 

1.8.6 Buoyancy 45 

1.8.7 Gas Laws 47 

1.8.8 Fluid viscosity 49 

1.8.9 Reynolds number 51 

1.8.10 Law of Continuity 53 

1.8.11 Viscous flow 54 

1.8.12 Bernoulli's equation 55 

1.8.13 Torricelli's equation 57 

1.8.14 Flow through a venturi tube 58 

Chemistry 61 

2.1 Terms and Definitions 61 

2.2 Periodic table 62 

2.3 Molecular quantities 63 

2.4 Stoichiometry 64 

2.5 Energy in chemical reactions 65 

2.6 Ions in liquid solutions 68 

2.7 pH 69 

DC electricity 73 

3.1 Electrical voltage 74 

3.2 Electrical current 79 

3.2.1 Electron versus conventional flow 82 

3.3 Electrical resistance and Ohm's Law 87 

3.4 Series versus parallel circuits 90 

3.5 Kirchhoff 's Laws 94 

3.6 Electrical sources and loads 98 

3.7 Resistors 99 

3.8 Bridge circuits 100 

3.8.1 Component measurement 101 

3.8.2 Sensor signal conditioning 103 

3.9 Capacitors 108 

3.10 Inductors 110 

AC electricity 113 

4.1 RMS quantities 114 

4.2 Resistance, Reactance, and Impedance 117 

4.3 Series and parallel circuits 117 

4.4 Phasor mathematics 118 

Introduction to Industrial Instrumentation 129 

5.1 Example: boiler water level control system 132 

5.2 Example: wastewater disinfection 137 

5.3 Example: chemical reactor temperature control 139 

5.4 Other types of instruments 141 


5.5 Summary 146 

6 Instrumentation documents 147 

6.1 Process Flow Diagrams 149 

6.2 Process and Instrument Diagrams 151 

6.3 Loop diagrams 153 

6.4 SAMA diagrams 156 

6.5 Instrument and process equipment symbols 160 

6.5.1 Line types 160 

6.5.2 Process/Instrument line connections 160 

6.5.3 Instrument bubbles 161 

6.5.4 Process valve types 162 

6.5.5 Valve actuator types 163 

6.5.6 Valve failure mode 164 

6.5.7 Flow measurement devices (flowing left-to-right) 165 

6.5.8 Process equipment 166 

6.5.9 SAMA diagram symbols 167 

7 Discrete process measurement 169 

7.1 "Normal" status of a switch 170 

7.2 Hand switches 172 

7.3 Limit switches 173 

7.4 Proximity switches 175 

7.5 Pressure switches 179 

7.6 Level switches 181 

7.7 Temperature switches 183 

7.8 Flow switches 185 

8 Analog electronic instrumentation 187 

8.1 4 to 20 mA analog current signals 187 

8.2 Relating 4 to 20 mA signals to instrument variables 190 

8.2.1 Example calculation: controller output to valve 190 

8.2.2 Example calculation: flow transmitter 191 

8.2.3 Example calculation: temperature transmitter 191 

8.2.4 Example calculation: pH transmitter 192 

8.2.5 Example calculation: reverse-acting I/P transducer signal 192 

8.2.6 Graphical interpretation of signal ranges 193 

8.3 Controller output current loops 196 

8.4 4- wire ("self-powered") transmitter current loops 198 

8.5 2-wire ("loop-powered") transmitter current loops 200 

8.6 Troubleshooting current loops 202 


9 Pneumatic instrumentation 209 

9.1 Pneumatic sensing elements 213 

9.2 Self-balancing pneumatic instrument principles 216 

9.3 Pilot valves and pneumatic amplifying relays 220 

9.4 Analogy to opamp circuits 228 

9.5 Analysis of a practical pneumatic instrument 237 

9.6 Proper care and feeding of pneumatic instruments 242 

9.7 Advantages and disadvantages of pneumatic instruments 243 

10 Digital electronic instrumentation 245 

10.1 The HART digital/analog hybrid standard 246 

10.1.1 HART multidrop mode 252 

10.1.2 HART multi-variable transmitters 253 

10.2 Fieldbus standards 254 

10.3 Wireless instrumentation 255 

11 Instrument calibration 257 

11.1 The meaning of calibration 257 

11.2 Zero and span adjustments (analog transmitters) 258 

11.3 LRV and URV settings, digital trim (digital transmitters) 261 

11.4 Calibration procedures 265 

11.4.1 Linear instruments 265 

11.4.2 Nonlinear instruments 265 

11.4.3 Discrete instruments 266 

11.5 Typical calibration errors 267 

11.5.1 As-found and as-left documentation 270 

11.5.2 Up-tests and Down-tests 270 

11.6 NIST traceability 271 

11.7 Instrument turndown 271 

11.8 Practical calibration standards 272 

11.8.1 Electrical standards 273 

11.8.2 Temperature standards 275 

11.8.3 Pressure standards 278 

11.8.4 Flow standards 284 

11.8.5 Analytical standards 285 

12 Continuous pressure measurement 289 

12.1 Manometers 290 

12.2 Mechanical pressure elements 295 

12.3 Electrical pressure elements 299 

12.3.1 Piezoresistive (strain gauge) sensors 300 

12.3.2 Differential capacitance sensors 303 

12.3.3 Resonant element sensors 308 

12.3.4 Mechanical adaptations 311 

12.4 Force-balance pressure transmitters 312 

12.5 Differential pressure transmitters 316 


12.6 Pressure sensor accessories 321 

12.6.1 Valve manifolds 322 

12.6.2 Bleed fittings 326 

12.6.3 Pressure pulsation dampening 327 

12.6.4 Remote and chemical seals 330 

12.6.5 Filled impulse lines 337 

12.6.6 Purged impulse lines 338 

12.6.7 Heat-traced impulse lines 340 

12.6.8 Water traps and pigtail siphons 342 

12.6.9 Mounting brackets 343 

12.7 Process/instrument suitability 344 

13 Continuous level measurement 347 

13.1 Level gauges (sightglasses) 348 

13.2 Float 352 

13.3 Hydrostatic pressure 357 

13.3.1 Bubbler systems 361 

13.3.2 Transmitter suppression and elevation 363 

13.3.3 Compensated leg systems 367 

13.3.4 Tank expert systems 372 

13.3.5 Hydrostatic interface level measurement 376 

13.4 Displacement 382 

13.4.1 Displacement interface level measurement 387 

13.5 Echo 389 

13.5.1 Ultrasonic level measurement 390 

13.5.2 Radar level measurement 395 

13.6 Laser level measurement 402 

13.7 Weight 403 

13.8 Capacitive 406 

13.9 Radiation 408 

13.10Level sensor accessories 409 

13.11Process/instrument suitability 412 

14 Continuous temperature measurement 415 

14.1 Bi-metal temperature sensors 417 

14.2 Filled-bulb temperature sensors 419 

14.3 Thermistors and Resistance Temperature Detectors (RTDs) 422 

14.4 Thermocouples 427 

14.5 Optical temperature sensing 435 

14.6 Temperature sensor accessories 437 

14.7 Process/instrument suitability 441 


15 Continuous fluid flow measurement 443 

15.1 Pressure-based flowmeters 444 

15.1.1 Venturi tubes and basic principles 449 

15.1.2 Orifice plates 458 

15.1.3 Other differential producers 467 

15.1.4 Proper installation 473 

15.1.5 High-accuracy flow measurement 477 

15.1.6 Equation summary 483 

15.2 Laminar flowmeters 485 

15.3 Variable-area flowmeters 487 

15.4 Velocity-based flowmeters 495 

15.4.1 Turbine flowmeters 496 

15.4.2 Vortex flowmeters 501 

15.4.3 Magnetic flowmeters 505 

15.4.4 Ultrasonic flowmeters 512 

15.5 Inertia-based (true mass) flowmeters 514 

15.5.1 Coriolis flowmeters 515 

15.6 Thermal-based (mass) flowmeters 524 

15.7 Positive displacement flowmeters 527 

15.8 Weighfeeders 528 

15.9 Change-of-quantity flow measurement 529 

15.10Insertion flowmeters 532 

15. 11 Process/instrument suitability 537 

16 Continuous analytical measurement 541 

16.1 Density measurement 541 

16.2 Turbidity measurement 541 

16.3 Conductivity measurement 542 

16.3.1 Dissociation and ionization in aqueous solutions 542 

16.3.2 Two-electrode conductivity probes 543 

16.3.3 Four-electrode conductivity probes 544 

16.3.4 Electrodeless conductivity probes 546 

16.4 pH measurement 549 

16.4.1 Colorimetric pH measurement 549 

16.4.2 Potentiometric pH measurement 550 

16.5 Chromatography 561 

17 Signal characterization 573 

17.1 Flow measurement in open channels 581 

17.2 Liquid volume measurement 583 

17.3 Radiative temperature measurement 591 

17.4 Analytical measurements 592 


18 Continuous feedback control 595 

18.1 Basic feedback control principles 596 

18.2 On/off control 602 

18.3 Proportional-only control 604 

18.4 Proportional-only offset 608 

18.5 Integral (reset) control 611 

18.6 Derivative (rate) control 614 

18.7 PID controller tuning 616 

A Doctor Strangeflow, or how I learned to relax and love Reynolds numbers 621 

B Creative Commons Attribution License 631 

B.l A simple explanation of your rights 632 

B.2 Legal code 633 



I did not want to write this book . . . honestly. 

My first book project began in 1998, titled Lessons In Electric Circuits, and I didn't call "quit" 
until six volumes and five years later. Even then, it was not complete, but being an open-source 
project it gained traction on the internet to the point where other people took over its development 
and it grew fine without me. The impetus for writing this first tome was a general dissatisfaction 
with available electronics textbooks. Plenty of textbooks exist to describe things, but few really 
explain things well for students, and the field of electronics is no exception. I wanted my book(s) 
to be different, and so they were. No one told me how time-consuming it was going to be to write 
them, though! 

The next few years' worth of my spare time went to developing a set of question-and-answer 
worksheets designed to teach electronics theory in a Socratic, active-engagement style. This project 
proved quite successful in my professional life as an instructor of electronics. In the summer of 2006, 
my job changed from teaching electronics to teaching industrial instrumentation, and I decided to 
continue the Socratic mode of instruction with another set of question-and-answer worksheets. 

However, the field of industrial instrumentation is not as well-represented as general electronics, 
and thus the array of available textbooks is not as vast. I began to re-discover the drudgery of 
trying to teach with inadequate texts as source material. The basis of my active teaching style was 
that students would spend time researching the material on their own, then engage in Socratic-style 
discussion with me on the subject matter when they arrived for class. This teaching technique 
functions in direct proportion to the quality and quantity of the source material at the students' 
disposal. Despite much searching, I was unable to find a textbook that adequately addressed my 
students' learning needs. Many textbooks I found were written in a shallow, "math-phobic" style 
that was well below the level I intended to teach to. Some reference books I found contained great 
information, but were often written for degreed engineers with lots of Laplace transforms and other 
mathematical techniques that were well above the level I intended to teach to. Few on either side of 
the spectrum actually made an effort to explain certain concepts that students generally struggle to 
understand. I needed a text that gave good, practical information and theoretical coverage at the 
same time. 

In a futile effort to provide my students with enough information to study outside of class, I 
scoured the internet for free tutorials written by others. While some manufacturer's tutorials were 
nearly perfect for my needs, others were just as shallow as the textbooks I had found, and/or were 
little more than sales brochures. I found myself starting to write my own tutorials on specific topics 
to "plug the gaps," but then another problem arose: it became troublesome for students to navigate 
through dozens of tutorials in an effort to find the information they needed in their studies. What 


my students really needed was a book, not a smorgasbord of tutorials. 

So here I am again, writing another textbook. This time around I have the advantage of wisdom 
gained from the first textbook project. For this project, I will not: 

«... attempt to maintain a parallel book in HTML markup (for direct viewing on the internet). 
I had to go to the trouble of inventing my own markup language last time in an effort to have 
multiple format versions of the book from the same source code. Instead, this time I will use 
stock P> the source code format and regular Adobe PDF format for the final output, 
which anyone may read thanks to its ubiquity. 

«... use a GNU GPL-style copyleft license. Instead, I will use the Creative Commons 
Attribution-only license, which makes things a lot easier for anyone wishing to incorporate my 
work into derivative works. My interest is maximum flexibility for those who may adapt my 
material to their own needs, not the imposition of certain philosophical ideals. 

«... start from a conceptual state of "ground zero." I will assume the reader has certain 
familiarity with electronics and mathematics, which I will build on. If a reader finds they need 
to learn more about electronics, they should go read Lessons In Electric Circuits. 

«... avoid using calculus to help explain certain concepts. Not all my readers will understand 
these parts, and so I will be sure to explain what I can without using calculus. However, 
I want to give my more mathematically adept students an opportunity to see the power of 
calculus applied to instrumentation where appropriate. By occasionally applying calculus and 
explaining my steps, I also hope this text will serve as a practical guide for students who might 
wish to learn calculus, so they can see its utility and function in a context that interests them. 

There do exist many fine references on the subject of industrial instrumentation. I only wish I 
could condense their best parts into a single volume for my students. Being able to do so would 
certainly save me from having to write my own! Listed here are some of the best books I can 
recommend for those wishing to explore instrumentation outside of my own presentation: 

• Handbook of Instrumentation and Controls, by Howard P. Kallen. Perhaps the best-written 
textbook on general instrumentation I have ever encountered. Too bad it's long out of print 
- my copy dates 1961. Like most American textbooks written during the years immediately 
following Sputnik, it is a masterpiece of practical content and conceptual clarity. 

• Industrial Instrumentation Fundamentals, by Austin E. Fribance. Another great post-Sputnik 
textbook - my copy dates 1962. 

• Instrumentation for Process Measurement and Control, by Normal A. Anderson. An inspiring 
effort by someone who knows the art of teaching as well as the craft of instrumentation. Too 
bad the content doesn't seem to have been updated since 1980. 

• Instrument Engineers' Handbook series (Volumes I, II, and III), edited by Bela Liptak. By far 
my favorite modern references on the subject. Unfortunately, there is a fair amount of material 
within that lies well beyond my students' grasp (Laplace transforms, etc.), and the volumes 
are incredibly bulky and expensive (1000+ pages, at a cost of nearly $200.00 apiece!). These 
texts also lack some of the basic content my students do need, and I don't have the heart to 
tell them to buy yet another textbook to fill the gaps. 


• Practically anything written by Francis Greg Shinskey. 

Whether or not I achieve my goal of writing a better textbook is a judgment left for others to 
make. One decided advantage my book will have over all the others is its openness. If you don't like 
anything you see in these pages, you have the right to modify it at will! Delete content, add content, 
modify content - it's all fair in this game we call "open source." My only condition is declared in the 
Creative Commons Attribution License: that you give me credit for my original authorship. What 
you do with it beyond that is wholly up to you. This way, perhaps I can spare someone else from 
having to write their own textbook from scratch! 


Chapter 1 



1.1 Terms and Definitions 

Mass (m) is the opposition that an object has to acceleration (changes in velocity). Weight is 
the force (F) imposed on a mass by a gravitational field. Mass is an intrinsic property of an 
object, regardless of the environment. Weight, on the other hand, depends on the strength of the 
gravitational field in which the object resides. A 20 kilogram slug of metal has the exact same mass 
whether it rests on Earth or in the zero-gravity environment of outer space. However, the weight 
of that mass depends on gravity: zero weight in outer space (where there is no gravity to act upon 
it), some weight on Earth, and a much greater amount of weight on the planet Jupiter (due to the 
much stronger gravitational field). 

Since mass is the opposition of an object to changes in velocity (acceleration), it stands to reason 
that force, mass, and acceleration for any particular object are directly related to one another: 

F = ma 


F = Force in newtons (metric) or pounds (British) 

m = Mass in kilograms (metric) or slugs (British) 

a = Acceleration in meters per second squared (metric) or feet per second squared (British) 

If the force in question is the weight of the object, then the acceleration (a) in question is the 
acceleration constant of the gravitational field where the object resides. For Earth at sea level, 
^gravity is approximately 9.8 meters per second squared, or 32 feet per second squared. Earth's 
gravitational acceleration constant is usually represented in equations by the variable letter g instead 
of the more generic a. 

Since acceleration is nothing more than the rate of velocity change with respect to time, the 
force/mass equation may be expressed using the calculus notation of the first derivative: 


i = m — 



F = Force in newtons (metric) or pounds (British) 

m = Mass in kilograms (metric) or slugs (British) 

v = Velocity in meters per second (metric) or feet per second (British) 

t = Time in seconds 

Since velocity is nothing more than the rate of position change with respect to time, the 
force/mass equation may be expressed using the calculus notation of the second derivative 
(acceleration being the derivative of velocity, which in turn is the derivative of position): 

d 2 x 

F = m — — 

dt 2 


F = Force in newtons (metric) or pounds (British) 
m = Mass in kilograms (metric) or slugs (British) 
x = Position in meters (metric) or feet (British) 
t = Time in seconds 



Mass density (p) for any substance is the proportion of mass to volume. Weight density (7) for 
any substance is the proportion of weight to volume. 

Just as weight and mass are related to each other by gravitational acceleration, weight density 
and mass density are also related to each other by gravity: 

weight TTig 

Weight and Mass 

7 = P9 

Weight density and Mass density 

1.2 Metric prefixes 


TGMk m p. n p 

tera giga mega kilo (none) milli micro nano pico 

10 12 10 9 10 6 10 3 10° icr 3 lcr 6 icr 9 icr 12 

10 2 10 1 icr 1 icr 2 
hecto deca deci centi 
h da d c 


1.3 Unit conversions and physical constants 

Converting between disparate units of measurement is the bane of many science students. The 
problem is worse for students of industrial instrumentation in the United States of America, who 
must work with British ("Customary") units such as the pound, the foot, the gallon, etc. World- 
wide adoption of the metric system would go a long way toward alleviating this problem, but until 
then it is important for students of instrumentation to master the art of unit conversions 1 . 

It is possible to convert from one unit of measurement to another by use of tables designed 
expressly for this purpose. Such tables usually have a column of units on the left-hand side and an 
identical row of units along the top, whereby one can look up the conversion factor to multiply by 
to convert from any listed unit to any other listed unit. While such tables are undeniably simple to 
use, they are practically impossible to memorize. 

The goal of this section is to provide you with a more powerful technique for unit conversion, 
which lends itself much better to memorization of conversion factors. This way, you will be able to 
convert between many common units of measurement while memorizing only a handful of essential 
conversion factors. 

I like to call this the unity fraction technique. It involves setting up the original quantity as 
a fraction, then multiplying by a series of fractions having physical values of unity (1) so that by 
multiplication the original value does not change, but the units do. Let's take for example the 
conversion of quarts into gallons, an example of a fluid volume conversion: 

35 qt = ??? gal 

Now, most people know there are four quarts in one gallon, and so it is tempting to simply 
divide the number 35 by four to arrive at the proper number of gallons. However, the purpose of 
this example is to show you how the technique of unity fractions works, not to get an answer to a 
problem. First, we set up the original quantity as a fraction, in this case a fraction with 1 as the 

35 qt 

Next, we multiply this fraction by another fraction having a physical value of unity, or 1. This 
means a fraction comprised of equal measures in the numerator and denominator, but with different 
units of measurement, arranged in such a way that the undesired unit cancels out leaving only the 
desired unit(s). In this particular example, we wish to cancel out quarts and end up with gallons, 
so we must arrange a fraction consisting of quarts and gallons having equal quantities in numerator 
and denominator, such that quarts will cancel and gallons will remain: 

35 qt\ /l gal 
^7 \~Ut 

x An interesting point to make here is that the United States did get something right when they designed their 
monetary system of dollars and cents. This is essentially a metric system of measurement, with 100 cents per 
dollar. The founders of the USA wisely decided to avoid the utterly confusing denominations of the British, with 
their pounds, pence, farthings, shillings, etc. The denominations of penny, dime, dollar, and eagle ($10 gold coin) 
comprised a simple power-of-ten system for money. Credit goes to France for first adopting a metric system of general 
weights and measures as their national standard. 


Now we see how the unit of "quarts" cancels from the numerator of the first fraction and the 
denominator of the second ("unity") fraction, leaving only the unit of "gallons" left standing: 

35 qt\ (I gal\ „ 

° 75 gal 

1 J V 4 qt 

The reason this conversion technique is so powerful is that it allows one to do a large range of 
unit conversions while memorizing the smallest possible set of conversion factors. 

Here is a set of six equal volumes, each one expressed in a different unit of measurement: 

1 gallon (gal) = 231.0 cubic inches (in 3 ) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) 
= 3.7854 liters (1) 

Since all six of these quantities are physically equal, it is possible to build a "unity fraction" out 
of any two, to use in converting any of the represented volume units into any of the other represented 
volume units. Shown here are a few different volume unit conversion problems, using unity fractions 
built only from these factors: 

40 gallons converted into fluid ounces: 

/40 gal\ /128 fl. oz 
V 1 / V Igal 

5120 fl. oz 

5.5 pints converted into cubic inches: 

5.5 pt\ /231 in 3 

1 ) \ 8 pt 

1170 liters converted into quarts: 

1170 1W 4 qt 

158.8 in 3 

1 J V 3.7854 1 

1236 qt 

By contrast, if we were to try to memorize a 6 x 6 table giving conversion factors between any 
two of six volume units, we would have to commit 30 different conversion factors to memory! Clearly, 
the ability to set up "unity fractions" is a much more memory-efficient and practical approach. 

But what if we wished to convert to a unit of volume measurement other than the six shown in 
the long equality? For instance, what if we wished to convert 5.5 pints into cubic feet instead of 
cubic inches? Since cubic feet is not a unit represented in the long string of quantities, what do we 

We do know of another equality between inches and feet, though. Everyone should know that 
there are 12 inches in 1 foot. All we need to do is set up another unity fraction in the original 
problem to convert cubic inches into cubic feet: 

5.5 pints converted into cubic feet (our first attempt!): 

'5.5 pt\ /231 in 3 \ / 1 ft 

1 ) V 8 pt ) \Y1 in 



Unfortunately, this will not give us the result we seek. Even though 1 ■ is a valid unity fraction, 
it does not completely cancel out the unit of inches. What we need is a unity fraction relating cubic 
feet to cubic inches. We can get this, though, simply by cubing the 1 ■ unity fraction: 

5.5 pints converted into cubic feet (our second attempt!): 

'5.5 pt\ /231 in 3 \ / 1 ft 
~1/ V 8 pt ) \12 in, 

Distributing the third power to the interior terms of the last unity fraction: 

'5.5 pt\ /231 in 3 \ / l 3 ft 3 

1 J V 8pt J Vl2 3 in 3 , 

Calculating the values of l 3 and 12 3 inside the last unity fraction, then canceling units and 

™J*)(?^)(J^\ = 0.0919 ft 3 
1 7 V § pt y Vl728in 3 / 

Once again, this unit conversion technique shows its power by minimizing the number of 
conversion factors we must memorize. We need not memorize how many cubic inches are in a 
cubic foot, or how many square inches are in a square foot, if we know how many linear inches are in 
a linear foot and we simply let the fractions "tell" us whether a power is needed for unit cancellation. 

A major caveat to this method of converting units is that the units must be directly proportional 
to one another, since this multiplicative conversion method is really nothing more than an exercise 
in mathematical proportions. Here are some examples (but not an exhaustive list!) of conversions 
that cannot be performed using the "unity fraction" method: 

• Absolute / Gauge pressures, because one scale is offset from the other by 14.7 PSI (atmospheric 

• Celsius / Fahrenheit, because one scale is offset from the other by 32 degrees. 

• Wire diameter / gauge number, because gauge numbers grow smaller as wire diameter grows 
larger (inverse proportion rather than direct) and because there is no proportion relating the 

• Power / decibels, because the relationship is logarithmic rather than proportional. 

The following subsections give sets of physically equal quantities, which may be used to create 
unity fractions for unit conversion problems. Note that only those quantities shown in the same line 
(separated by = symbols) are truly equal to each other, not quantities appearing in different lines! 


1.3.1 Conversion formulae for temperature 

• °F = (°C)(9/5) + 32 

• °C = (°F - 32) (5/9) 

• °R = °F + 459.67 

• K = °C + 273.15 

1.3.2 Conversion factors for distance 

1 inch (in) = 2.540000 centimeter (cm) 
1 foot (ft) = 12 inches (in) 
1 yard (yd) = 3 feet (ft) 
1 mile (mi) = 5280 feet (ft) 

1.3.3 Conversion factors for volume 

1 gallon (gal) = 231.0 cubic inches (in 3 ) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) 
= 3.7854 liters (1) 

1 milliliter (ml) = 1 cubic centimeter (cm 3 ) 

1.3.4 Conversion factors for velocity 

1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934 
kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot - international) 

1.3.5 Conversion factors for mass 

1 pound (lbm) = 0.45359 kilogram (kg) = 0.031081 slugs 

1.3.6 Conversion factors for force 

1 pound-force (lbf) = 4.44822 newton (N) 

1.3.7 Conversion factors for area 

1 acre = 43560 square feet (ft 2 ) = 4840 square yards (yd 2 ) = 4046.86 square meters (m 2 ) 

1.3.8 Conversion factors for pressure (either all gauge or all absolute) 

1 pound per square inch (PSI) = 2.03603 inches of mercury (in. Hg) = 27.6807 inches of water (in. 
W.C.) = 6.894757 kilo-pascals (kPa) 


1.3.9 Conversion factors for pressure (absolute pressure units only) 

1 atmosphere (Atm) = 14.7 pounds per square inch absolute (PSIA) = 760 millimeters of mercury 
absolute (mmHgA) = 760 torr (torr) = 1.01325 bar (bar) 

1.3.10 Conversion factors for energy or work 

1 British thermal unit (Btu - "International Table") = 251.996 calories (cal - "International Table") 
= 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071 watt-hour (W-hr) = 1.05506 x 10 10 
ergs (erg) = 778.169 foot-pound-force (ft-lbf) 

1.3.11 Conversion factors for power 

1 horsepower (hp - 550 ft-lbf/s) = 745.7 watts (W) = 2544.43 British thermal units per hour 
(Btu/hr) = 0.0760181 boiler horsepower (hp - boiler) 

1.3.12 Terrestrial constants 

Acceleration of gravity at sea level = 9.806650 meters per second per second (m/s 2 ) = 32.1740 feet 
per second per second (ft/s 2 ) 

Atmospheric pressure = 14.7 pounds per square inch absolute (PSIA) = 760 millimeters of mercury 
absolute (mmHgA) = 760 torr (torr) = 1.01325 bar (bar) 

Atmospheric gas concentrations: 

• Nitrogen = 78.084 % 

• Oxygen = 20.946 % 

• Argon = 0.934 % 

• Carbon Dioxide (C0 2 ) = 0.033 % 

• Neon = 18.18 ppm 

• Helium = 5.24 ppm 

• Methane (CH 4 ) = 2 ppm 

• Krypton =1.14 ppm 

• Hydrogen = 0.5 ppm 

• Nitrous Oxide (N 2 0) = 0.5 ppm 

• Xenon = 0.087 ppm 


1.3.13 Properties of water 

Freezing point at sea level = 32°F = 0°C 

Boiling point at sea level = 212°F = 100°C 

Density of water at 4°C = 1000 kg/m 3 = 1 g/cm 3 = 1 kg/liter = 62.428 lb/ft 3 = 1.951 slugs/ft 3 

Specific heat of water at 14°C = 1.00002 calories/g-°C = 1 BTU/lb-°F = 4.1869 joules/g-°C 

Specific heat of ice w 0.5 calories/g-°C 

Specific heat of steam w 0.48 calories/g-°C 

Absolute viscosity of water at 20°C = 1.0019 centipoise (cp) = 0.0010019 Pascal-seconds (Pa-s) 

Surface tension of water (in contact with air) at 18°C = 73.05 dynes/cm 

pH of pure water at 25° C = 7.0 (pH scale = to 14) 

1.3.14 Properties of dry air at sea level 

Density of dry air at 20°C and 760 torr = 1.204 mg/cm 3 = 1.204 kg/m 3 = 0.075 lb/ft 3 = 0.00235 
slugs/ft 3 

Absolute viscosity of dry air at 20°C and 760 torr = 0.018 centipoise (cp) = 1.8 x 10~ 5 Pascal- 
seconds (Pa-s) 

1.3.15 Miscellaneous physical constants 

Speed of light in a vacuum (c) = 2.9979 x 10 8 meters per second (m/s) = 186,281 miles per second 


Avogadro's number (Na) = 6.0220 x 10 23 per mole (mol -1 ) 

Electronic charge (e) = 1.6022 x 10~ 19 Coulomb (C) 

Faraday constant (F) = 9.6485 x 10 4 Coulombs per mole (C/mol) 

Boltzmann's constant (k) = 1.3807 x 10~ 23 joules per Kelvin (J/K) 

Stefan-Boltzmann constant (er) = 5.6703 x 10~ 8 Watts per square meter-Kelvin 4 (W/m 2 -K 4 ) 

Molar gas constant (R) = 8.3144 joules per mole-Kelvin (J/mol-K) 

Note: all physical constants listed here were derived (rounded to the fifth significant digit) from 
values given on page F-198 of the CRC Handbook of Chemistry and Physics, 64th edition. 


1.3.16 Weight densities of common materials 

All density figures approximate for samples at standard temperature and pressure. 



Gasoline: 7 = 41 lb/ft 3 to 43 lb/ft 

• Naphtha, petroleum: 7 = 41.5 lb/ft 3 

• Acetone: 7 = 49.4 lb/ft 3 

• Ethanol (ethyl alcohol): 7 = 49.4 lb/ft 3 






Methanol (methyl alcohol): 7 = 50.5 lb/ft 


Kerosene: 7 = 51.2 lb/ft 


Toluene: 7 = 54.1 lb/ft 


Benzene: 7 = 56.1 lb/ft 
Olive oil: 7 = 57.3 lb/ft 



• Coconut oil: 7 = 57.7 lb/ft 








Linseed oil (boiled): 7 = 58.8 lb/ft 


Castor oil: 7 = 60.5 lb/ft 


Sea water: 7 = 63.99 lb/ft 


Milk: 7 = 64.2 lb/ft 3 to 64.6 lb/ft 3 
Ethylene glycol (ethanediol): 7 = 69.22 lb/ft 


Glycerin: 7 = 78.6 lb/ft 


• Mercury: 7 = 849 lb/ft 3 


• Balsa wood: 7 = 7 lb/ft 3 to 9 lb/ft 3 

• Cork: 7 = 14 lb/ft 3 to 16 lb/ft 3 



Maple wood: 7 = 39 lb/ft 3 to 47 lb/ft 

Ice: 7 = 57.2 lb/ft 


• Tar: 7 = 66 lb/ft 3 

• Rubber (soft): 7 = 69 lb/ft 3 

• Rubber (hard): 7 = 74 lb/ft 3 


• Calcium: 7 = 96.763 lb/ft 
















Sugar: 7 = 99 lb/ft 


Magnesium: 7 = 108.50 lb/ft 


Beryllium: 7 = 115.37 lb/ft 


Rock salt: 7 = 136 lb/ft 


Quartz: 7 = 165 lb/ft 


Cement (set): 7 = 170 lb/ft 3 to 190 lb/ft 

Carbon (diamond): 7 = 196.65 lb/ft 3 to 220.37 lb/ft 

Chromium: 7 = 448.86 lb/ft 


Iron: 7 = 490.68 lb/ft 
Brass: 7 = 524.4 lb/ft 



Copper: 7 = 559.36 lb/ft 


Molybdenum: 7 = 638.01 lb/ft 


Lead: 7 = 708.56 lb/ft 
Gold: 7 = 1178.6 lb/ft 





1.4 Dimensional analysis 

An interesting parallel to the "unity fraction" unit conversion technique is something referred to in 
physics as dimensional analysis. Performing dimensional analysis on a physics formula means to set 
it up with units of measurement in place of variables, to see how units cancel and combine to form 
the appropriate unit(s) of measurement for the result. 

For example, let's take the familiar power formula used to calculate power in a simple DC electric 

P = IV 


P = Power (watts) 
I = Current (amperes) 
V = Voltage (volts) 

Each of the units of measurement in the above formula (watt, ampere, volt) are actually 
comprised of more fundamental physical units. One watt of power is one joule of energy transferred 
per second. One ampere of current is one coulomb of electric charge moving by per second. One 
volt of potential is one joule of energy per coulomb of electric charge. When we write the equation 
showing these units in their proper orientations, we see that the result (power in watts, or joules 
per second) actually does agree with the units for amperes and volts because the unit of electric 
charge (coulombs) cancels out. In dimensional analysis we customarily distinguish unit symbols 
from variables by using non-italicized letters and surrounding each one with square brackets: 

P = IV 

[Watts] = [Amperes] x [Volts] or [W] = [A][V] 











Dimensional analysis gives us a way to "check our work" when setting up new formulae for 
physics- and chemistry-type problems. 



1.5 The International System of Units 

The very purpose of physics is to quantitatively describe and explain the physical world in as few 
terms as possible. This principle extends to units of measurement as well, which is why we usually 
find different units used in science actually defined in terms of more fundamental units. The watt, 
for example, is one joule of energy transferred per second of time. The joule, in turn, is defined in 
terms of three base units, the kilogram, the meter, and the second: 



[ S 2] 

Within the metric system of measurements, an international standard exists for which units 
are considered fundamental and which are considered "derived" from the fundamental units. The 
modern standard is called SI, which stands for Systeme International. This standard recognizes 
seven fundamental, or base units, from which all others are derived 2 : 

Physical quantity 

SI unit 

SI symbol 










Electric current 






Amount of substance 



Luminous intensity 



An older standard existed for base units, in which the centimeter, gram, and second comprised 
the first three base units. This standard is referred to as the cgs system, in contrast to the SI 
system 3 . You will still encounter some derived cgs units used in instrumentation, including the poise 
and the stokes (both used to express fluid viscosity) . Then of course we have the British engineering 
system which uses such wonderful units as feet, pounds, and (thankfully) seconds. Despite the fact 
that the majority of the world uses the metric (SI) system for weights and measures, the British 
system is sometimes referred to as the Customary system. 

2 The only exception to this rule being units of measurement for angles, over which there has not yet been full 
agreement whether the unit of the radian (and its solid counterpart, the steradian) is a base unit or a derived unit. 
3 The older name for the SI system was "MKS," representing meters, kilograms, and seconds. 
4 I'm noting my sarcasm here, just in case you are immune to my odd sense of humor. 


1.6 Conservation Laws 

The Law of Mass Conservation states that matter can neither be created nor destroyed. The Law of 
Energy Conservation states that energy can neither be created nor destroyed. However, both mass 
and energy may change forms, and even change into one another in the case of nuclear phenomena. 

Conversion of mass into energy, or of energy into mass, is quantitatively described by Albert 
Einstein's famous equation: 

E = mc 2 


E = Energy (joules) 

m = Mass (kilograms) 

c = Speed of light (approximately 3 x 10 8 meters per second) 

1.7 Classical mechanics 

Classical mechanics (often called Newtonian mechanics in honor of Isaac Newton) deal with forces 
and motions of objects in common circumstances. The vast majority of instrumentation applications 
deals with this realm of physics. Two other areas of physics, relativistic and quantum, will not 
be covered in this chapter because their domains lie outside the typical experience of industrial 
instrumentation 5 . 

5 Relativistic physics deals with phenomena arising as objects travel near the velocity of light. Quantum physics 
deals with phenomena at the atomic level. Neither is germane to the vast majority of industrial instrument 


1.7.1 Newton's Laws of Motion 

These laws were formulated by the great mathematician and physicist Isaac Newton (1642-1727). 
Much of Newton's thought was inspired by the work of an individual who died the same year Newton 
was born, Galileo Galilei (1564-1642). 

1. An object at rest tends to stay at rest; an object in motion tends to stay in motion 

2. The acceleration of an object is directly proportional to the net force acting upon it and 
inversely proportional to the object's mass 

3. Forces between objects always exist in equal and opposite pairs 

Newton's first law may be thought of as the law of inertia, because it describes the property of 
inertia that all objects having mass exhibit: resistance to change in velocity. 

Newton's second law is the verbal equivalent of the force/mass/acceleration formula: F = ma 

Newton's third law describes how forces always exist in pairs between two objects. The rotating 
blades of a helicopter, for example, exert a downward force on the air (accelerating the air) , but the 
air in turn exerts an upward force on the helicopter (suspending it in flight). These two forces are 
equal in magnitude but opposite in direction. Such is always the case when forces exist between 


1.7.2 Work and Energy 

Work is the expenditure of energy resulting from exerting a force over a parallel displacement 
(motion) 6 : 

W = Fx 


W = Work, in joules (metric) or foot-pounds (English) 

F = Force doing the work, in newtons (metric) or pounds (English) 

X = Displacement over which the work was done, in meters (metric) or feet (English) 

Potential energy is energy existing in a stored state, having the potential to do useful work. If we 
perform work in lifting a mass vertically against the pull of earth's gravity, we store potential energy 
which may later be released by allowing the mass to return to its previous altitude. The equation 
for potential energy in this case is just a special form of the work equation (W = Fx), where work is 
now expressed as potential energy (W = E p ), force is now expressed as a weight caused by gravity 
acting on a mass (F = mg), and displacement is now expressed as a height (x = h): 

W = Fx 

E p = mgh 


E p = Potential energy in joules (metric) or foot-pounds (British) 

to = Mass of object in kilograms (metric) or slugs (British) 

g = Acceleration of gravity in meters per second squared (metric) or feet per second squared 

h = Height of lift in meters (metric) or feet (British) 

Kinetic energy is energy in motion. The kinetic energy of a moving mass is equal to: 

E k = -mv 


Ek = Potential energy in joules (metric) or foot-pounds (British) 

m = Mass of object in kilograms (metric) or slugs (British) 

v = Velocity of mass in meters per second (metric) or feet per second (British) 

The Law of Energy Conservation is extremely useful in projectile mechanics problems, where 
we typically assume a projectile loses no energy and gains no energy in its flight. The velocity of 

6 Technically, the best way to express work resulting from force and displacement is in the form of a vector dot- 
product: W = F ■ x. The result of a dot product is always a scalar quantity (neither work nor energy possesses a 
direction, so it cannot be a vector), and the result is the same magnitude as a scalar product only if the two vectors 
are pointed in the same direction. 


a projectile, therefore, depends on its height above the ground, because the sum of potential and 
kinetic energies must remain constant: 

Ep + Ek = constant 

In free-fall problems, where the only source of energy for a projectile is its initial height, the 
initial potential energy must be equal to the final kinetic energy: 

E p (initial) = Ek (final) 

We can see from this equation that mass cancels out of both sides, leaving us with this simpler 

gh t = -v f 

It also leads to the paradoxical conclusion that the mass of a free-falling object is irrelevant to 
its velocity. That is, both a heavy object and a light object in free fall will hit the ground with 
the same velocity, and fall for the same amount of time, if released from the same height under the 
influence of the same gravity 7 . 

Dimensional analysis confirms the common nature of energy whether in the form of potential, 
kinetic, or even mass (as described by Einstein's equation). First, we will set these three energy 
equations next to each other for comparison of their variables: 

Potential energy due to elevation 

Kinetic energy due to velocity 

Mass-to-energy equivalence 

Next, we will dimensionally analyze them using standard SI metric units (kilogram, meter, 
second). Following the SI convention, mass (m) is always expressed in kilograms [kg], distance (h) 
in meters [m], and time (t) in seconds [s]. This means velocity (t>, or c for the velocity of light) in 
the SI system will be expressed in meters per second [m/s] and acceleration (a, or g for gravitational 
acceleration) in meters per second squared [m/s 2 ]: 


= mgh 


1 2 

= -mv 

E = mc 


rml 2 
[kg] — Kinetic energy due to velocity 

-t^t- — = [kg] — [m] Potential energy due to elevation 

[s^J Ls^J 

[kg][m 2 ] rml 2 

7 In practice, we usually see heavy objects fall faster than light objects due to the resistance of air. Energy losses 
due to air friction nullify our assumption of constant total energy during free-fall. Energy lost due to air friction never 
translates to velocity, and so the heavier object ends up hitting the ground faster (and sooner) because it had much 
more energy than the light object did to start. 


[kg] [m 2 1 , fin 

rmi z 
[kg] — Mass-to-energy equivalence 

[s*] l OJ L s 

In all three cases, the unit for energy is the same: kilogram-meter squared per second squared. 
This is the fundamental definition of a "joule" of energy, and it is the same result given by all three 


1.7.3 Mechanical springs 

Many instruments make use of springs to translate force into motion, or visa- versa. The basic "Ohm's 
Law" equation for a mechanical spring relating applied force to spring motion (displacement) is called 
Hooke's Law 8 : 

F = -kx 


F = Force generated by the spring in newtons (metric) or pounds (English) 

k = Constant of elasticity, or "spring constant" in newtons per meter (metric) or pounds per 

foot (English) 

x = Displacement of spring in meters (metric) or feet (English) 

Hooke's Law is a linear function, just like Ohm's Law is a linear function: doubling the 
displacement (either tension or compression) doubles the spring's force. At least this is how springs 
behave when they are displaced a small percentage of their total length. If you displace a spring 
more substantially, the spring material will become strained beyond its elastic limit and either yield 
(permanently deform) or fail (break). 

The amount of potential energy stored in a tensed spring may be predicted using calculus. We 
know that potential energy stored in a spring is the same as the amount of work done on the spring, 
and work is equal to the product of force and displacement (assuming parallel lines of action for 

E p = Fx 

Thus, the amount of work done on a spring is the force applied to the spring (F = kx) multiplied 
by the displacement (x). The problem is, the force applied to a spring varies with displacement 
and therefore is not constant as we compress or stretch the spring. Thus, in order to calculate the 
amount of potential energy stored in the spring (E p = Fx), we must calculate the amount of energy 
stored over infinitesimal amounts of displacement (Fdx, or kxdx) and then add those bits of energy 
up (J) to arrive at a total: 

E p = kx dx 
We may evaluate this integral using the power rule (x is raised to the power of 1 in the integrand) : 

1 9 

E p = -kx 2 + E 


Ep = Energy stored in the spring in joules (metric) or foot-pounds (English) 

k = Constant of elasticity, or "spring constant" in newtons per meter (metric) or pounds per 

foot (English) 

8 Hooke's Law may be written as F = kx without the negative sign, in which case the force (F) is the force applied 
on the spring from an external source. Here, the negative sign represents the spring's reaction force to being displaced 
(the restoring force). A spring's reaction force always opposes the direction of displacement: compress a spring, and 
it pushes back on you; stretch a spring, and it pulls back. A negative sign is the mathematically symbolic way of 
expressing the opposing direction of a vector. 



x = Displacement of spring in meters (metric) or feet (English) 

Eq = The constant of integration, representing the amount of energy initially stored in the spring 
prior to our displacement of it 

For example, if we take a very large spring with a constant k equal to 60 pounds per foot and 
displace it by 4 feet, we will store 480 foot-pounds of potential energy in that spring (i.e. we will do 
480 foot-pounds of work on the spring) . 

Graphing the force-displacement function on a graph yields a straight line (as we would expect, 
because Hooke's Law is a linear function). The area accumulated underneath this line from feet 
to 4 feet represents the integration of that function over the interval of to 4 feet, and thus the 
amount of potential energy stored in the spring: 

(pounds) 200 











/ Work = 480 foot-pounds 

Displacement (x) 

Note how the geometric interpretation of the shaded area on the graph exactly equals the result 
predicted by the equation E p = ^kx 2 : the area of a triangle is one-half times the base times the 
height. One-half times 4 feet times 240 pounds is 480 foot-pounds. 


1.8 Fluid mechanics 

A fluid is any substance having the ability to flow: to freely change shape and move under the 
influence of a motivating force. Fluid motion may be analyzed on a microscopic level, treating each 
fluid molecule as an individual projectile body. This approach can be extraordinarily tedious on a 
practical level, but still useful as a simple model of fluid motion. 

Some fluid properties are accurately predicted by this model, especially predictions dealing with 
potential and kinetic energies. However, the ability of a fluid's molecules to independently move give 
it unique properties that solids do not possess. One of these properties is the ability to effortlessly 
transfer pressure, defined as force applied over area. 



1.8.1 Pressure 

The common phases of matter are solid, liquid, and gas. Liquids and gases are fundamentally distinct 
from solids in their intrinsic inability to maintain a fixed shape. In other words, liquids and gases 
tend to fill whatever solid containers they are held in. Similarly, both liquids and gases both have 
the ability to flow, which is why they are collectively called fluids. 

Due to their lack of definite shape, fluids tend to disperse any force applied to them. This stands 
in marked contrast to solids, which tend to transfer force with the direction unchanged. Take for 
example the force transferred by a nail, from a hammer to a piece of wood: 

on nail 




on wood 

The impact of the hammer's blow is directed straight through the solid nail into the wood below. 
Nothing surprising here. But now consider what a fluid would do when subjected to the same 
hammer blow: 



on I exerted 

\^L on piston 



on cylinder 




Given the freedom of a fluid's molecules to move about, the impact of the hammer blow becomes 
directed everywhere against the inside surface of the container (the cylinder). This is true for all 
fluids: liquids and gases alike. The only difference between the behavior of a liquid and a gas in the 
same scenario is that the gas will compress (i.e. the piston will move down as the hammer struck 
it), whereas the liquid will not compress (i.e. the piston will remain in its resting position). Gases 
yield under pressure, liquids do not. 

It is very useful to quantify force applied to a fluid in terms of force per unit area, since the force 
applied to a fluid becomes evenly dispersed in all directions to the surface containing it. This is the 
definition of pressure (P): how much force (F) is distributed across how much area (^4). 



In the metric system, the standard unit of pressure is the Pascal (Pa), defined as one Newton 
(N) of force per square meter (m 2 ) of area. In the English system of measurement, the standard unit 
of pressure is the PSI: pounds (lb) of force per square inch (in 2 ) of area. Pressure is often expressed 
in units of kilo-pascals (kPa) when metric units are used because one pascal is a rather low pressure 
in most engineering applications. 

The even distribution of force throughout a fluid has some very practical applications. One 
application of this principle is the hydraulic lift, which functions somewhat like a fluid lever: 




Hydraulic lift 







Lever and fulcrum 



v / / / / 


Force applied to the small piston creates a pressure throughout the fluid. That pressure exerts 
a greater force on the large piston than what is exerted on the small piston, by a factor equal to 
the ratio of piston areas. If the large piston has five times the area of the small piston, force will be 
multiplied by five. Just like with the lever, however, there must be a trade-off so we do not violate 
the Conservation of Energy. The trade-off for increased force is decreased distance, whether in the 
lever system or in the hydraulic lift system. If the large piston generates a force five times greater 
than what was input at the small piston, it will move only one-fifth the distance that the small 
piston does. In this way, energy in equals energy out (remember that work, which is equivalent to 
energy, is calculated by multiplying force by parallel distance traveled). 

For those familiar with electricity, what you see here in either the lever system or the hydraulic 
lift is analogous to a transformer: we can step AC voltage up, but only by reducing AC current. 
Being a passive device, a transformer cannot boost power. Therefore, power out can never be greater 
than power in, and given a perfectly efficient transformer, power out will always be precisely equal 
to power in: 

Power = (Voltage in) (Current in) = (Voltage out) (Current out) 

Work = (Force in) (Distance in) = (Force out) (Distance out) 



Fluid may be used to transfer power just as electricity is used to transfer power. Such systems 
are called hydraulic if the fluid is a liquid (usually oil) , and pneumatic if the fluid is a gas (usually 
air). In either case, a machine (pump or compressor) is used to generate a continuous fluid pressure, 
pipes are used to transfer the pressurized fluid to the point of use, and then the fluid is allowed to 
exert a force against a piston or a set of pistons to do mechanical work: 

Hydraulic power system 






Pneumatic power system 






An interesting use of fluid we see in the field of instrumentation is as a signaling medium, to 
transfer information between places rather than to transfer power between places. This is analogous 
to using electricity to transmit voice signals in telephone systems, or digital data between computers 
along copper wire. Here, fluid pressure represents some other quantity, and the principle of force 
being distributed equally throughout the fluid is exploited to transmit that representation to some 
distant location, through piping or tubing: 


Closed bulb 

filled with 





This illustration shows a simple temperature-measuring system called a filled bulb, where an 
enclosed bulb filled with fluid is exposed to a temperature that we wish to measure. Heat causes the 
fluid pressure to increase, which is sent to the gauge far away through the pipe, and registered at 
the gauge. The purpose of the fluid here is two-fold: first to sense temperature, and second to relay 
this temperature measurement a long distance away to the gauge. The principle of even pressure 
distribution allows the fluid to act as a signal medium to convey the information (bulb temperature) 
to a distant location. 



1.8.2 Pascal's Principle and hydrostatic pressure 

We learned earlier that fluids tend to evenly distribute the force applied to them. This tendency is 
known as Pascal's principle, and it is the fundamental principle upon which fluid power and fluid 
signaling systems function. In the example of a hydraulic lift given earlier, we assume that the 
pressure throughout the fluid pathway is equal: 

Pressure = 



(150 lbs) 


Hydraulic lift 

(3 in 2 ) 

Pressure = 

(27 in 2 ) 


(1350 lbs) 




The key assumption we make here is that the only force we need to consider on the fluid is the 
force exerted on the small piston (150 pounds). If this is truly the only force acting on the fluid, 
then it will likewise be the only source of fluid pressure, and pressure will simply be equal to force 
divided by area (150 pounds -f- 3 square inches = 50 PSI). 

However, when we are dealing with tall columns of fluid, and/or dense fluids, there is another 
force we must consider: the weight of the fluid itself. Suppose we took a cubic foot of water which 
weighs approximately 62.4 pounds, and poured it into a tall, vertical tube with a cross-sectional 
area of 1 square inch: 



tube area = 1 in 2 

Water column 
weight = 62.4 lbs 

Pressure gauge 
l_L JJ /J^-x 

62.4 PSI 

Naturally, we would expect the pressure measured at the bottom of this tall tube to be 62.4 
pounds per square inch, since the entire column of water (weighing 62.4 pounds) has its weight 
supported by one square inch of area. 

If we placed another pressure gauge mid-way up the tube, though, how much pressure would it 
register? At first you might be inclined to say 62.4 PSI as well, because you learned earlier in this 
lesson that fluids naturally distribute force throughout their bulk. However, in this case the pressure 
is not the same mid-way up the column as it is at the bottom: 


■A r > 

(Half-way up) 

tube area = 1 in 2 

Water column 
weight = 62.4 lbs 

Pressure gauge 


Pressure gauge 

62.4 PSI 

The reason for this apparent discrepancy is that the source of pressure in this fluid system comes 
from the weight of the water column itself. Half-way up the column, the water only experiences half 
the total weight (31.2 pounds), and so the pressure is half of what it is at the very bottom. We never 
dealt with this effect before, because we assumed the force exerted by the piston in the hydraulic 
lift was so large that it "swamped" the weight of the fluid itself. Here, with our very tall column 
of water (144 feet tall!), the effect of gravity upon the water's mass is quite substantial. Indeed, 
without a piston to exert an external force on the water, weight is the only source of force we have 
to consider when calculating pressure. 

An interesting fact about pressure generated by a column of fluid is that the width or shape of 
the containing vessel is irrelevant: the height of the fluid column is the only dimension we need to 
consider. Examine the following tube shapes, all connected at the bottom: 




Since the force of fluid weight is generated only along the axis of gravitational attraction (straight 
down), that is the only axis of measurement important in determining "hydrostatic" fluid pressure. 

The fixed relationship between the vertical height of a water column and pressure is such that 
sometimes water column height is used as a unit of measurement for pressure. That is, instead of 
saying "30 PSI," we could just as correctly quantify that same pressure as 830.4 inches of water 
("W.C. or "H2O), the conversion factor being approximately 27.68 inches of vertical water column 
per PSI. 

As one might guess, the density of the fluid in a vertical column has a significant impact on 
the hydrostatic pressure that column generates. A liquid twice as dense as water, for example, will 
produce twice the pressure for a given column height. For example, a column of this liquid (twice 
as dense as water) 14 inches high will produce a pressure at the bottom equal to 28 inches of water 
(28 "W.C), or just over 1 PSI. An extreme example is liquid mercury, which is over 13.5 times as 
dense as water. Due to its exceptional density and ready availability, the height of a mercury column 
is also used as a standard unit of pressure measurement. For instance, 25 PSI could be expressed 
as 50.9 inches of mercury ("Hg), the conversion factor being approximately 2.036 inches of vertical 
mercury column per PSI. 

The mathematical relationship between vertical liquid height and hydrostatic pressure is quite 
simple, and may be expressed by either of the following formulae: 

P = pgh 

P = jh 


P = Hydrostatic pressure in units of weight per square area unit: Pascals (N/m 2 ) or lb/ft 2 

p = Mass density of liquid in kilograms per cubic meter (metric) or slugs per cubic foot (British) 

g = Acceleration of gravity (9.8 meters per second squared or 32 feet per second squared) 

7 = Weight density of liquid in newtons per cubic meter (metric) or pounds per cubic foot 


h = Vertical height of liquid column 

Dimensional analysis vindicates these formulae in their calculation of hydrostatic pressure. 
Taking the second formula as an example: 

P = jh 






As you can see, the unit of "feet" in the height term cancels out one of the "feet" units in the 
denominator of the density term, leaving an answer for pressure in units of pounds per square foot. 
If one wished to set up the problem so that the answer presented in a more common pressure unit 
such as pounds per square inch, both the liquid density and height would have to be expressed in 
appropriate units (pounds per cubic inch and inches, respectively). 


Applying this to a realistic problem, consider the case of a tank filled with 8 feet (vertical) of 
castor oil, having a weight density of 60.5 pounds per cubic foot. This is how we would set up the 
formula to calculate for hydrostatic pressure at the bottom of the tank: 

/60.5 lb\ , „ , 

p -h?-) (8ft) 


484 lb 

If we wished to convert this result into a more common unit such as PSI (pounds per square 
inch), we could do so using an appropriate fraction of conversion units: 

484 lb\ / 1 ft 2 

ft 2 J V 144 in 2 

P = 5^ = 3.36 PSI 


1.8.3 Fluid density expressions 

Fluid density is commonly expressed as a ratio in comparison to pure water at standard 
temperature 9 . This ratio is known as specific gravity. For example, the specific gravity of glycerin 
may be determined by dividing the density of glycerin by the density of water: 

Specific gravity of any liquid — 




Specific gravity of glycerin = ^ ycerm __ _^^ = 1.26 

Dglycerin _ 78.6 lb/ft 
D wa ter 62.4 lb/ft 3 

As with all ratios, specific gravity is a unitless quantity. Note how the identical units of pounds 
per cubic foot cancel out of both numerator and denominator, to leave a quotient with no unit at 

Industry-specific units of measurement do exist for expressing the relative density of a fluid. These 
units of measurement all begin with the word "degree" much the same as for units of temperature 
measurement. They are as follows: 

The mathematical relationships between each of these "degree" units of density versus specific 
gravity 10 is as follows: 


Degrees API = 131.5 

Specific gravity 

Degrees Twaddell = 200 x (Specific gravity — 1) 

Two different formulae exist for the calculation of degrees Baume, depending on whether the 
liquid in question is heavier or lighter than water. For lighter-than-water liquids: 


Degrees Baume (light) = 130 

Specific gravity 

Note that pure water would measure 10° Baume on the light scale. As liquid density decreases, 
the light Baume value increases. For heavier-than-water liquids: 


Degrees Baume (heavy) = 145 — 

Specific gravity 

Note that pure water would measure 0° Baume on the heavy scale. As liquid density increases, 
the heavy Baume value increases. Just to make things confusing, there are different standards for the 
heavy Baume scale. Instead of the constant value 145 shown in the above equation (used throughout 
the United States of America), an older Dutch standard used the same formula with a constant value 
of 144. The Gerlach heavy Baume scale uses a constant value of 146.78: 

Degrees Baume (heavy, old Dutch) = 144 

Specific gravity 

9 Usually, this standard temperature is 4 degrees Celsius, the point of maximum density for water. However, 
sometimes the specific gravity of a fluid will be expressed in relation to the density of water at some other temperature. 
10 For each of these calculations, specific gravity is defined as the ratio of the liquid's density at 60 degrees Fahrenheit 
to the density of pure water, also at 60 degrees Fahrenheit. 


Degrees Baume (heavy, Gerlach scale) = 146.78 

Specific gravity 

There exists a seemingly endless array of "degree" scales used to express liquid density, scattered 
throughout the pages of history. For the measurement of sugar concentrations in the food industries, 
the unit of degrees Balling was invented. This scale was later revised to become the unit of degrees 
Brix, which directly corresponds to the percent concentration of sugar in the liquid. The density of 
tanning liquor may be measured in degrees Bark. Milk density may be measured in degrees Soxhlet. 
Vegetable oil density (and in older times, the density of oil extracted from sperm whales) may be 
measured in degrees Oleo. 



1.8.4 Manometers 

Expressing fluid pressure in terms of a vertical liquid column makes perfect sense when we use a very 
simple kind of motion-balance pressure instrument called a manometer. A manometer is nothing 
more than a piece of clear (glass or plastic) tubing filled with a liquid of known density, situated 
next to a scale for measuring distance. The most basic form of manometer is the U-tube manometer, 
shown here: 

U-tube manometer 





h Height 

Pressure is read on the scale as the difference in height (h) between the two liquid columns. One 
nice feature of a manometer is that it really cannot become "uncalibrated" so long as the fluid is 
pure and the assembly is maintained in an upright position. If the fluid used is water, the manometer 
may be filled and emptied at will, and even rolled up for storage if the tubes are made of flexible 

We may build even more sensitive manometers by purposely inclining one or more of the tubes, 
so that distance read along the tube length is a fractional proportion of distance measured along 
the vertical: 

Inclined manometer 



This way, a greater motion of liquid is required to generate the same hydrostatic pressure (vertical 
liquid displacement) than in an upright manometer, making the inclined manometer more sensitive. 

If even more sensitivity is desired, we may build something called a micromanometer, consisting 
of a gas bubble trapped in a clear horizontal tube between two large vertical manometer chambers: 

A simple micromanometer 



Pressure applied to the top of either vertical chamber will cause the vertical liquid columns to 
shift just the same as any U-tube manometer. However, the bubble trapped in the clear horizontal 
tube will move much further than the vertical displacement of either liquid column, owing to the 
huge difference in cross-sectional area between the vertical chambers and the horizontal tube. This 
amplification of motion makes the micromanometer exceptionally sensitive to small pressures. 

A common form of manometer seen in calibration laboratories is the well type, consisting of a 
single vertical tube and a relatively large reservoir (called the "well") acting as the second column: 

"Well" manometer 



= Scale 

Due to the well's much larger cross-sectional area, liquid motion inside of it is negligible compared 
to the motion of liquid inside the clear viewing tube. For all practical purposes, the only liquid motion 
is inside the smaller tube. Thus, the well manometer provides an easier means of reading pressure: 


no longer does one have to measure the difference of height between two liquid columns, only the 
height of a single column. 


1.8.5 Systems of pressure measurement 

Pressure measurement is often a relative thing. What we mean when we say there is 35 PSI of air 
pressure in an inflated car tire is that the pressure inside the tire is 35 pounds per square inch greater 
than the surrounding, ambient air pressure. It is a fact that we live and breathe in a pressurized 
environment. Just as a vertical column of liquid generates a hydrostatic pressure, so does a vertical 
column of gas. If the column of gas is very tall, the pressure generated by it will be substantial 
enough to measure. Such is the case with Earth's atmosphere, the pressure at sea level caused by 
the weight of the atmosphere is approximately 14.7 PSI. 

You and I do not perceive this constant air pressure around us because the pressure inside our 
bodies is equal to the pressure outside our bodies. Thus our skin, which serves as a differential 
pressure-sensing diaphragm, detects no difference of pressure between the inside and outside of our 
bodies. The only time the Earth's air pressure becomes perceptible to us is if we rapidly ascend or 
descend in a vehicle, where the pressure inside our bodies does not have time to equalize with the 
pressure outside, and we feel the force of that differential pressure on our eardrums. 

If we wish to speak of a fluid pressure in terms of how it compares to a perfect vacuum (absolute 
zero pressure), we specify it in terms of absolute units. For example, when I said earlier that the 
atmospheric pressure at sea level was 14.7 PSI, what I really meant is that it is 14.7 PSIA (pounds 
per square inch absolute), meaning 14.7 pounds per square inch greater than a perfect vacuum. 
When I said earlier that the air pressure inside an inflated car tire was 35 PSI, what I really meant 
is that it was 35 PSIG (pounds per square inch gauge), meaning 35 pounds per square inch greater 
than ambient air pressure. When units of pressure measurement are specified without a "G" or "A" 
suffix, it is usually (but not always!) assumed that gauge pressure (relative to ambient pressure) is 

This offset of 14.7 PSI between absolute and gauge pressures can be confusing if we must convert 
between different pressure units. Suppose we wished to express the tire pressure of 35 PSIG in 
units of inches of water column ("W.C.). If we stay in the gauge-pressure scale, all we have to do is 
multiply by 27.68: 

35 PSI 27.68 "W.C. „„ T ^ 

— x ^psi— = 968 ' 8 wc " 

Note how the fractions have been arranged to facilitate cancellation of units. The "PSI" unit 
in the numerator of the first fraction cancels with the "PSI" unit in the denominator of the second 
fraction, leaving inches of water column ("W.C.) as the only unit standing. Multiplying the first 
fraction (35 PSI over 1) by the second fraction (27.68 "W.C. over 1 PSI) is "legal" to do since the 
second fraction has a physical value of unity (1): being that 27.68 inches of water column is the 
same physical pressure as 1 PSI, the second fraction is really the number "1" in disguise. As we 
know, multiplying any quantity by unity does not change its value, so the result of 968.8 "W.C. we 
get has the exact same physical meaning as the original figure of 35 PSI. 

If, however, we wished to express the car's tire pressure in terms of inches of water column 
absolute (in reference to a perfect vacuum), we would have to include the 14.7 PSI offset in our 
calculation, and do the conversion in two steps: 

35 PSIG + 14.7 PSI = 49.7 PSIA 


49.7 PSIA 27.68 "W.C.A 
1 1 PSIA 

The proportion between inches of water column and pounds per square inch is still the same 
(27.68) in the absolute scale as it is in the gauge scale. The only difference is that we included the 
14.7 PSI offset in the very beginning to express the tire's pressure on the absolute scale rather than 
on the gauge scale. From then on, all conversions were in absolute units. 

There are some pressure units that are always in absolute terms. One is the unit of atmospheres, 1 
atmosphere being 14.7 PSIA. There is no such thing as "atmospheres gauge" pressure. For example, 
if we were given a pressure as being 4.5 atmospheres and we wanted to convert that into pounds per 
square inch gauge (PSIG), the conversion would be a two-step process: 

4.5 atm 14.7 PSIA ^ nT A 

x = 66.15 PSIA 

1 1 atm 

66.15 PSIA - 14.7 PSI = 51.45 PSIG 

Another unit of pressure measurement that is always absolute is the torr, equal to 1 millimeter 
of mercury column absolute (mmHgA). torr is absolute zero, equal to atmospheres, PSIA, or 
-14.7 PSIG. Atmospheric pressure at sea level is 760 torr, equal to 1 atmosphere, 14.7 PSIA, or 

If we wished to convert the car tire's pressure of 35 PSIG into torr, we would once again have to 
offset the initial value to get everything into absolute terms. 

35 PSIG + 14.7 PSI = 49.7 PSIA 

49.7 PSIA 760 torr 

—i x i4Tps!A =2569 ' 5torr 


4 r > 

1.8.6 Buoyancy 

When a solid body is immersed in a fluid, it displaces an equal volume of that fluid. This displacement 
of fluid generates an upward force on the object called the buoyant force. The magnitude of this 
force is equal to the weight of the fluid displaced by the solid body, and it is always directed exactly 
opposite the line of gravitational attraction. This is known as Archimedes' Principle. 

Buoyant force is what makes ships float. A ship sinks into the water just enough so that the 
weight of the water displaced is equal to the total weight of the ship and all it holds (cargo, crew, 
food, fuel, etc.): 

Amount of water 
displaced by the ship 

If we could somehow measure the weight of that water displaced, we would find it exactly equals 
the dry weight of the ship: 





Archimedes' Principle also explains why hot-air balloons and helium aircraft float. By filling a 
large enclosure with a gas that is less dense than the surrounding air, that enclosure experiences 
an upward (buoyant) force equal to the difference between the weight of the air displaced and the 
weight of the gas enclosed. If this buoyant force equals the weight of the craft and all it holds (cargo, 
crew, food, fuel, etc.), it will exhibit an apparent weight of zero, which means it will float. If the 
buoyant force exceeds the weight of the craft, the resultant force will cause an upward acceleration 
according to Newton's Second Law of motion (F = ma). 

Submarines also make use of Archimedes' Principle, adjusting their buoyancy by adjusting the 
amount of water held by ballast tanks on the hull. Positive buoyancy is achieved by "blowing" water 
out of the ballast tanks with high-pressure compressed air, so that the submarine weighs less (but 
still occupies the same hull volume and therefore displaces the same amount of water). Negative 
buoyancy is achieved by "flooding" the ballast tanks so that the submarine weighs more. Neutral 
buoyancy is when the buoyant force exactly equals the weight of the submarine and the remaining 


water stored in the ballast tanks, so that the submarine is able to "hover" in the water with no 
vertical acceleration or deceleration. 

An interesting application of Archimedes' Principle is the quantitative determination of an 
object's density by submersion in a liquid. For instance, copper is 8.96 times as dense as water, 
with a mass of 8.96 grams per cubic centimeter (8.96 g/cm 3 ) as opposed to water at 1.00 gram per 
cubic centimeter (1.00 g/cm 3 ). If we had a sample of pure, solid copper exactly 1 cubic centimeter 
in volume, it would have a mass of 8.96 grams. Completely submerged in pure water, this same 
sample of solid copper would appear to have a mass of only 7.96 grams, because it would experience 
a buoyant force equivalent to the mass of water it displaces (1 cubic centimeter = 1 gram of water). 
Thus, we see that the difference between the dry mass (mass measured in air) and the wet mass 
(mass measured when completely submerged in water) is the mass of the water displaced. Dividing 
the sample's dry mass by this mass difference (dry — wet mass) yields the ratio between the sample's 
mass and the mass of an equivalent volume of water, which is the very definition of specific gravity. 
The same calculation yields a quantity for specific gravity if weights instead of masses are used, 
since weight is nothing more than mass multiplied by the acceleration of gravity (F we i g } lt = mg), 
and the constant g cancels out of both numerator and denominator: 

c .„ „ m dry m dry g Dry weight 

bpecinc Gravity — — — 

mdry ~ m wet m dry g - m wet g Dry weight - Wet weight 


1.8.7 Gas Laws 

The Ideal Gas Law relates pressure, volume, molecular quantity, and temperature of an ideal gas 
together in one neat mathematical expression: 

PV = nRT 


P = Absolute pressure (atmospheres) 

V = Volume (liters) 

n = Gas quantity (moles) 

R = Universal gas constant (0.0821 L • atm / mol • K) 

T = Absolute temperature (K) 

An alternative form of the Ideal Gas Law uses the number of actual gas molecules (N) instead 
of the number of moles of molecules (n) : 

PV = NkT 


P = Absolute pressure (atmospheres) 

V = Volume (liters) 

N = Gas quantity (moles) 

k = Boltzmann's constant (1.38 x 10~ 23 J / K) 

T = Absolute temperature (K) 

Although no gas in real life is ideal, the Ideal Gas Law is a close approximation for conditions of 
modest gas density, and no phase changes (gas turning into liquid or visa- versa). 

Since the molecular quantity of an enclosed gas is constant, and the universal gas constant must 
be constant, the Ideal Gas Law may be written as a proportionality instead of an equation: 

PV ocT 
Several "gas laws" are derived from this Ideal Gas Law. They are as follows: 

PV = Constant Boyle's Law (assuming constant temperature T) 

V oc T Charles's Law (assuming constant pressure P) 

P oc T Gay-Lussac's Law (assuming constant volume V) 

You will see these laws referenced in explanations where the specified quantity is constant (or 
very nearly constant). 


For non-ideal conditions, the "Real" Gas Law formula incorporates a corrected term for the 
compressibility of the gas: 

PV = ZnRT 


P = Absolute pressure (atmospheres) 

V = Volume (liters) 

Z = Gas compressibility factor (unitless) 

n = Gas quantity (moles) 

R = Universal gas constant (0.0821 L • atm / mol • K) 

T = Absolute temperature (K) 

The compressibility factor for an ideal gas is unity (Z = 1), making the Ideal Gas Law a limiting 
case of the Real Gas Law. Real gases have compressibility factors less than unity (< 1). 


1.8.8 Fluid viscosity 

Viscosity is a measure of a fluid's internal friction. The more "viscous" a fluid is, the "thicker" it is 
when stirred. Clean water is an example of a low-viscosity liquid, while honey at room temperature 
is an example of a high- viscosity liquid. 

There are two different ways to quantify the viscosity of a fluid: absolute viscosity and kinematic 
viscosity. Absolute viscosity (symbolized by the Greek symbol "eta" n, or sometimes by the Greek 
symbol "mu" /i), also known as dynamic viscosity, is a direct relation between stress placed on 
a fluid and its rate of deformation (or shear). The textbook definition of absolute viscosity is 
based on a model of two flat plates moving past each other with a film of fluid separating them. 
The relationship between the shear stress applied to this fluid film (force divided by area) and the 
velocity/film thickness ratio is viscosity: 


F ^^^- plate Velocity 

t I I — -v 


J X / ////////////////// ///////// ' < stationar y> 




n = Absolute viscosity (pascal-seconds) 

F = Force (newtons) 

L = Film thickness (meters) - typically much less than 1 meter for any realistic demonstration! 

A = Plate area (square meters) 

v = Relative velocity (meters per second) 

Another common unit of measurement for absolute viscosity is the poise, with 1 poise being equal 
to 0.1 pascal-seconds. Both units are too large for common use, and so absolute viscosity is often 
expressed in centipoise. Water has an absolute viscosity of very nearly 1.000 centipoise. 

Kinematic viscosity (symbolized by the Greek letter "nu" v) includes an assessment of the fluid's 
density in addition to all the above factors. It is calculated as the quotient of absolute viscosity and 
mass density: 


v = — 



v = Kinematic viscosity (stokes) 

rj = Absolute viscosity (poises) 

p = Mass density (grams per cubic centimeter) 


As with the unit of poise, the unit of stokes is too large for convenient use, so kinematic viscosities 
are often expressed in units of centistokes. Water has an absolute viscosity of very nearly 1.000 

The mechanism of viscosity in liquids is inter-molecular cohesion. Since this cohesive force is 
overcome with increasing temperature, most liquids tend to become "thinner" (less viscous) as they 
heat up. The mechanism of viscosity in gases, however, is inter-molecular collisions. Since these 
collisions increase in frequency and intensity with increasing temperature, gases tend to become 
"thicker" (more viscous) as they heat up. 

As a ratio of stress to strain (applied force to yielding velocity), viscosity is often constant for 
a given fluid at a given temperature. Interesting exceptions exist, though. Fluids whose viscosities 
change with applied stress, and/or over time with all other factors constant, are referred to as non- 
Newtonian fluids. A simple example of a non-Newtonian fluid is cornstarch mixed with water, which 
"solidifies" under increasing stress then returns to a liquid state when the stress is removed. 


1.8.9 Reynolds number 

Viscous flow is when friction forces dominate the behavior of a moving fluid, typically in cases where 
viscosity (internal fluid friction) is great. Inviscid flow, by contrast, is where friction within a moving 
fluid is negligible. The Reynolds number of a fluid is a dimensionless quantity expressing the ratio 
between a moving fluid's momentum and its viscosity. 

A couple of formulae for calculating Reynolds number of a flow are shown here: 



Re = Reynolds number (unitless) 

D = Diameter of pipe, (meters) 

V = Average velocity of fluid (meters per second) 

p = Mass density of fluid (kilograms per cubic meter) 

fi = Absolute viscosity of fluid (Pascal-seconds) 

= (3160)G f Q 


Re = Reynolds number (unitless) 

Gf = Specific gravity of liquid (unitless) 

Q = Flow rate (gallons per minute) 

D = Diameter of pipe (inches) 

/i = Absolute viscosity of fluid (centipoise) 

The Reynolds number of a fluid stream may be used to qualitatively predict whether the flow 
regime will be laminar or turbulent. Low Reynolds number values predict laminar flow, where fluid 
molecules move in straight "stream-line" paths, and fluid velocity near the center of the pipe is 
substantially greater than near the pipe walls: 

Laminar flow 

pipe wall 

Fluid flow ^^ =^ "profile" 

High Reynolds number values predict turbulent flow, where individual molecule motion is chaotic 
on a microscopic scale, and fluid velocities across the face of the flow profile are similar: 


Turbulent flow 

pipe wall 

nuid flow ^^^v^^p ) y el0 <?y 

pipe wall 

A generally accepted rule-of-thumb is that Reynolds number values less than 10,000 will probably 
be laminar, while values in excess of 10,000 will probably be turbulent. There is no definite threshold 
value for all fluids and piping configurations, though. 



1.8.10 Law of Continuity 

Any fluid moving through a pipe obeys the Law of Continuity, which states that the product of 
average velocity (v) , pipe cross-sectional area (A) , and fluid density (p) for a given flow stream must 
remain constant: 


Fluid continuity is an expression of a more fundamental law of physics: the Conservation of 
Mass. If we assign appropriate units of measurement to the variables in the continuity equation, we 
see that the units cancel in such a way that only units of mass per unit time remain: 


This means that in order for the product pAv to differ between any two points in a pipe, 
mass would have to mysteriously appear and disappear. So long as the pipe does not leak, this 
is impossible without violating the Law of Mass Conservation. The continuity principle for fluid 
through a pipe is analogous to the principle of current being the same everywhere in a series circuit, 
and for equivalently the same reason. 

We refer to a fluid as incompressible if its density does not substantially change. For this limiting 
case, the continuity equation simplifies to the following form: 


m 3 

r 2 " 
m z 

1 _ 




A1V1 = A 2 v 2 

The practical implication of this principle is that fluid velocity is inversely proportional to the 
cross-sectional area of a pipe. That is, fluid slows down when the pipe's diameter expands, and 
visa-versa. We see this principle easily in nature: deep rivers run slow, while rapids are relatively 
shallow (and/or narrow). 


1.8.11 Viscous flow 

The pressure dropped by a slow-moving, viscous fluid through a pipe is described by the Hagen- 
Poiseuille equation. This equation applies only for conditions of low Reynolds number; i.e. when 
viscous forces are the dominant restraint to fluid motion through the pipe, and turbulence is 


Q = Flow rate (gallons per minute) 

k = Unit conversion factor = 7.86 xlO 5 

AP = Pressure drop (inches of water column) 

D = Pipe diameter (inches) 

/i = Liquid viscosity (centipoise) - this is a temperature-dependent variable! 

L = Length of pipe section (inches) 


1.8.12 Bernoulli's equation 

Bernoulli's equation is an expression of the Law of Energy Conservation for an inviscid fluid stream, 
named after Daniel Bernoulli 11 . It states that the sum total energy at any point in a passive fluid 
stream (i.e. no pumps or other energy-imparting machines in the flow path) must be constant. Two 
versions of the equation are shown here: 


•-1W- -^ + f'i = <>./>g + ^Y + P2 

25 7 

z = Height of fluid (from a common reference point, usually ground level) 

p = Mass density of fluid 

7 = Weight density of fluid (7 = pg) 

g = Acceleration of gravity 

V = Velocity of fluid 

P = Pressure of fluid 

Each of the three terms in Bernoulli's equation is an expression of a different kind of energy, 
commonly referred to as head: 

zpg Elevation head 


+ P\ 



z\ + — H = zi 

v p 


Velocity head 

Pressure head 

Elevation and Pressure heads are potential forms of energy, while Velocity head is a kinetic form 
of energy. Note how the elevation and velocity head terms so closely resemble the formulae for 
potential and kinetic energy of solid objects: 

E p = mgh Potential energy formula 

Ek = -mv Kinetic energy formula 

It is very important to maintain consistent units of measurement when using Bernoulli's equation! 
Each of the three energy terms (elevation, velocity, and pressure) must possess the exact same units 
if they are to add appropriately 12 . Here is an example of dimensional analysis applied to the first 
version of Bernoulli's equation (using British units): 

11 According to Ven Te Chow in Open Channel Hydraulics, who quotes from Hunter Rouse and Simon Ince's work 
History of Hydraulics, Bernoulli's equation was first formulated by the great mathematician Leonhard Euler and made 
popular by Julius Weisbach, not by Daniel Bernoulli himself. 

12 Surely you've heard the expression, "Apples and Oranges don't add up." Well, pounds per square inch and 
pounds per square foot don't add up either! 





v z p 










ft 3 

s 2 


ft 3 

ft 2 

ft-s 2 


As you can see, both the first and second terms of the equation (elevation and velocity heads) 
bear the same unit of slugs per foot-second squared after all the "feet" are canceled. The third term 
(pressure head) does not appear as though its units agree with the other two terms, until you realize 
that the unit definition of a "pound" is a slug of mass multiplied by the acceleration of gravity in 
feet per second squared, following Newton's Second Law of motion (F = ma): 

[lb] = [slug] 

Once we make this substitution into the pressure head term, the units are revealed to be the 
same as the other two terms, slugs per foot-second squared: 



ft 2 

ft-s 2 

In order for our British units to be consistent here, we must use feet for elevation, slugs per 
cubic foot for mass density, feet per second squared for acceleration, feet per second for velocity, 
and pounds per square foot for pressure. If one wished to use the more common pressure unit of 
PSI (pounds per square inch) with Bernoulli's equation instead of PSF (pounds per square foot), 
all the other units would have to change accordingly: elevation in inches, mass density in slugs per 
cubic inch, acceleration in inches per second squared, and velocity in inches per second. 

Just for fun, we can try dimensional analysis on the second version of Bernoulli's equation, this 
time using metric units: 

v 2 P 

2g 7 



[ N " 
m 2 


m 3 

Here, we see that all three terms end up being cast in simple units of meters. That is, the fluid's 
elevation, velocity, and pressure heads are all expressed as simple elevations. In order for our metric 
units to be consistent here, we must use meters for elevation, meters per second for velocity, meters 
per second squared for acceleration, pascals (newtons per square meter) for pressure, and newtons 
per cubic meter for weight density. 



1.8.13 Torricelli's equation 

The velocity of a liquid stream exiting from a nozzle, pressured solely by a vertical column of that 
same liquid, is equal to the free-fall velocity of a solid mass dropped from the same height as the 
top of the liquid column. In both cases, potential energy (in the form of vertical height) converts to 
kinetic energy (motion): 

o Mass 

(same velocities) 

Bernoulli's more 
v after setting the 

This was discovered by Evangelista Torricelli almost 100 years prior to 
comprehensive formulation. The velocity may be determined by solving for 
potential and kinetic energy formulae equal to each other (since all potential energy at the upper 
height must translate into kinetic energy at the bottom, assuming no frictional losses): 




2gh = v 2 

Note how mass {in) simply disappears from the equation, neatly canceling on both sides. This 
means the nozzle velocity depends only on height, not the mass density of the liquid. It also means 
the velocity of the falling object depends only on height, not the mass of the object. 


1.8.14 Flow through a venturi tube 

If an incompressible fluid moves through a venturi tube (a tube purposefully built to be narrow in 
the middle), the continuity principle tells us the fluid velocity must increase through the narrow 
portion. This increase in velocity causes kinetic energy to increase at that point. If the tube is 
level with the earth, there is negligible difference in elevation (z) between different points of the 
tube's centerline, which means elevation head remains constant. According to the Law of Energy 
Conservation, some other form of energy must decrease to account for the increase in kinetic energy. 
This other form is the pressure head, which decreases at the throat of the venturi: 



(less than upstream) 

Ideally, the pressure downstream of the narrow throat should be the same as the pressure 
upstream, assuming equal pipe diameters upstream and down. However, in practice the downstream 
pressure gauge will show slightly less pressure than the upstream gauge due to some inevitable energy 
loss as the fluid passed through the venturi. Some of this loss is due to fluid friction against the 
walls of the tube, and some is due to viscous losses within the fluid driven by turbulent fluid motion 
at the high-velocity throat passage. 

The difference between upstream and downstream pressure is called permanent pressure loss, 
while the difference in pressure between the narrow throat and downstream is called pressure 

If we install vertical sight-tubes called piezometers along a horizontal venturi tube, the differences 
in pressure will be shown by the heights of liquid columns within the tubes. Here, we assume an 
ideal (inviscid) liquid with no permanent pressure loss: 





Ground level 

If we add three more piezometers to the venturi tube assembly, each one equipped with its own 
Pitot tube facing upstream to "catch" the velocity of the fluid, we see that total energy is indeed 
conserved at every point in the system. Here, each of the "heads" represented in Bernoulli's equation 
are shown in relation to the different piezometer heights: 

v 2 P . 

z H 1 = (constant) 

2.9 7 


Zj Z2 Z3 

I I I 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 / 1 1 1 1 1 1 1 1 



A more realistic scenario would show the influence of energy lost in the system due to friction. 
Here, the total energy is seen to decrease as a result of friction: 

v//2g ' 

energy line 




Chow, Ven Te., Open-Channel Hydraulics, McGraw-Hill Book Company, Inc., New York, NY, 1959. 

Giancoli, Douglas C, Physics for Scientists & Engineers, Third Edition, Prentice Hall, Upper Saddle 
River, New Jersey, 2000. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Miller, Richard W., Flow Measurement Engineering Handbook, Second Edition, McGraw-Hill 
Publishing Company, New York, NY, 1989. 

Rouse, Hunter, Characteristics of Laminar and Turbulent Flow (video), Iowa Institute of Hydraulic 
Research, University of Iowa. 

Shapiro, Ascher H., Pressure Fields and Fluid Acceleration (video), Massachusetts Institute of 
Technology, Educational Services Incorporated, 1962. 

Vennard, John K., Elementary Fluid Mechanics, 3rd Edition, John Wiley & Sons, Inc., New York, 
NY, 1954. 

Weast, Robert C; Astel, Melvin J.; and Beyer, William H., CRC Handbook of Chemistry and 
Physics, 64th Edition, CRC Press, Inc., Boca Raton, FL, 1984. 

Chapter 2 


2.1 Terms and Definitions 

• Atom: the smallest unit of matter that may be isolated by chemical means. 

• Element: a substance composed of atoms all sharing the same number of protons in their 

• Particle: a part of an atom, separable from the other portions only by levels of energy far in 
excess of chemical reactions. 

• Molecule: the smallest unit of matter composed of two or more atoms joined by electron 
interaction in a fixed ratio. The smallest unit of a compound. 

• Ion: an atom or molecule that is not electrically balanced. 

• Compound: a substance composed of identical molecules. 

• Mixture: a substance composed of different atoms or molecules. 




2.2 Periodic table 


Periodic Table of the Elements 

He 2 







Li 3 
2a 1 

Be 4 

2s 2 

Name ""~ 

K 19 
. Potassium 
39.0983 * 

— Atomic ma 
(averaged a 

B 5 


C 6 
2p 2 

N 7 

2p 3 

O 8 

F 9 
2p 6 

Ne 10 

2p 6 


Mg 12 


Al 13 

Si 14 

P 15 

S 16 

CI 17 

Ar 18 


3s 2 


3p 2 

3p 3 

3p 4 

3p 6 

3p 6 

K 19 


Ca 20 


So 21 


Ti 22 


V 23 

Cr 24 

Mn 25 


Fe 26 


Co 27 


Ni 28 


Cu 29 

Zn 30 


Ga 31 


Gs 32 



Aa 33 


Se 34 


Br 35 

Kr 36 


4s 2 

3d 1 4s 2 

3d 2 4s 2 

3d 3 4s 2 

Sd^s 1 

3d s 4s 2 

3d E 4s 2 

3d 7 4s 2 

3d B 4s 2 

3d ID 4s' 

3d 10 4s 2 


4p 2 

4p 3 


4p 6 

4p G 

Rb 37 


Sr 38 


Y 39 

Zr 40 
91 .224 

Nb 41 

Mo 42 


Tc 43 

Ru 44 



Rh 45 

Pd 46 


Ag 47 


Cd 48 


In 49 


Sn 50 


Sb 51 


Te 52 



I 53 


Xa 54 


5s 1 

5s 2 

4d'5s 2 

4d 2 5s 2 

4d 4 5s 1 

4d 5 5s' 

4d 5 5s 2 

4d 7 5s 1 

4d s 5s 1 

4d 1D 5s° 


4d 1D 5s 2 


5p 2 

5p 3 


5p 6 

5p 6 

Cs 55 


Ba 56 




Hf 72 

Ta 73 


W 74 

Re 75 
1 86.207 

Ob 76 



lr 77 

Pt 78 


Au 79 


Hg 80 


Pb 82 


Bi 83 

Po 84 



At 85 



Rn 86 



6s 2 

5d 2 6s 2 

5d 3 6s 2 

5d'6s 2 

5d 5 6s 2 

5d E 6s 2 

5d 7 6s 2 

5d a 6s' 


5d 10 6s 2 


6p 2 

6p 3 


6p 5 

6p 6 

Fr 87 


7s 1 

Ra 88 

7s 2 


Unq 1 04 

6d 2 7s 2 

Unp 105 

6d 3 7s 2 

Unh 106 

6d 1 7s 2 

Uns 107 



La 57 


Ce 58 

4f'5d'6s 2 

Pr 59 

4f 3 6s 2 

Nd 60 



4t'6s E 

Pm 61 



4f 5 6s 2 

Sm 62 

1 50.36 
4f s 6s 2 

Eu 63 

4f 7 6s 2 

Gd 64 
4! 7 5d'6s 2 

Tb 65 

4f a 6s 2 

Dy 66 


1 62.50 

4f ID 6s 2 

Ho 67 

4t"6s 2 

Er 68 
4f l2 6s 2 

Tm 69 

4t' 3 6s 2 

Yb 70 
4f 1, 6s 2 

Lu 71 


4f IJ 5d'6s 2 

Ac 89 

Th 90 


6b 1 7b 2 

6d 2 7s 2 

U 92 

Np 93 

Pu 94 






5f 3 6d'7s 2 

sfedVs 2 

5f s 6d°7s 2 



5f 7 6d'7s 2 


n 100 

Md 101 

No 102 

Lr 103 







2 6d°7s 2 

51 l3 6d'-'7s 2 

6d°7s 2 

6d'7s 2 

Attributes of each element may be interpreted in each table entry as such. In this example, we 
have the element Potassium: 

K 19 


4s 1 

The atomic number (number of protons in the nucleus of each Potassium atom) is 19. This 
number defines the element. If we were to somehow to add or subtract protons from the nucleus of 
a Potassium atom, it would cease being Potassium and transmutate into a different element. 

The atomic mass or atomic weight (combined number of protons and neutron in the nucleus 
of each Potassium atom) is 39. Neutrons may be added to or taken away from an atom's nucleus 
without changing its elemental identity. Atoms with the same number of protons but different 
numbers of neutrons in the nucleus are called isotopes. Isotopes have the same chemical properties, 
but may have different nuclear properties (such as stability - whether or not the atom is likely to 
spontaneously decay, which we refer to as radioactivity). The periodic table entry shows an atomic 
mass of slightly more than 39 for Potassium because different isotopes of Potassium exist in nature. 
The table's entries for atomic mass reflect the relative abundances of each element's isotopes as 
naturally found on the earth. Individually, though, the atomic mass of a single atom will always be 
a whole number (just like the atomic number). 

The outer- most electron shell configuration is shown here as 4s 1 , telling us that a neutral 
Potassium atom has 1 electron residing in the "s" subshell of the 4th shell. The configuration of an 


atom's electrons in the outermost different shells and subshells determines its chemical properties 
(i.e. its tendency to bond with other atoms to form molecules). 

2.3 Molecular quantities 

Sample sizes of chemical substances are often measured in moles. One mole of a substance is defined 
as a sample having 6.022 x 10 23 (Avogadro's number) molecules 1 . An elemental sample's mass is 
equal to its molecular quantity in moles multiplied by the element's atomic mass in amu (atomic 
mass units). For example, 2.00 moles of naturally-occurring Potassium will have a mass of 78.2 

When referring to liquid solutions, the concentration of a solute is often expressed as a molarity, 
defined as the number of moles of solute per liter of solution. Molarity is usually symbolized by an 
italicized capital letter M. It is important to bear in mind that the volume used to calculate molarity 
is that of the total solution (solute plus solvent) and not the solvent alone. 

Suppose we had a solution of salt-water, comprised of 33.1 grams of table salt thoroughly mixed 
with pure water to make a total volume of 1.39 liters. In order to calculate the molarity of this 
solution, we first need to determine the equivalence between moles of salt and grams of salt. Since 
table salt is sodium chloride (NaCl), and we know the atomic masses of both sodium (23.0 amu) 
and chlorine (35.5 amu), we may easily calculate the mass of one mole of salt: 

1 mole of NaCl = 23.0 g + 35.5 g = 58.5 g 

We may use this equivalence as a unity fraction to help us convert the number of grams of salt 
per unit volume of solution into a molarity (moles of salt molecules per liter) : 

33.1 g\ / 1 mol \ mol 

w ' 0.407 = 0.407 M 

1.39 1/ V58.5 g 1 

1 Truth be told, a "mole" is 6.022 X 10 23 of literally any discrete entities. There is nothing wrong with measuring 
the amount of eggs in the world using the unit of the mole. Think of "mole" as a really big dozen! 



2.4 Stoichiometry 

Stoichiometry is the balancing of atoms in a chemical equation. It is an expression of the Law of 
Mass Conservation, in that elements are neither created nor destroyed in a chemical reaction. Thus, 
the numbers and types of atoms in a reaction product sample must be the same as the numbers and 
types of atoms in the reactants which reacted to produce it. For example: 

CH 4 + 20 2 -> C0 2 + 2H 2 


Reaction products 

Carbon = 1x1 

Carbon =1x1 

Hydrogen = 1x4 

Hydrogen = 2x2 

Oxygen = 2x2 

Oxygen = (1 x 2) + (2 x 1) 

As you can see in this example, every single atom entering the reaction is accounted for in the 
reaction products. The only exception to this rule is in nuclear reactions where elements transmutate. 
No such transmutation occurs in any mere chemical reaction, and so we may safely assume equal 
numbers and types of atoms before and after any chemical reaction. Chemical reactions strictly 
involve re-organization of molecular bonds, with electrons as the constituent particles comprising 
those bonds. Nuclear reactions involve the re-organization of atomic nuclei (protons, neutrons, etc.), 
with far greater energy levels associated. 


2.5 Energy in chemical reactions 

A chemical reaction that results in the net release of energy is called exothermic. Conversely, a 
chemical reaction that requires a net input of energy to occur is called endothermic. The relationship 
between chemical reactions and energy exchange is correlated with the breaking or making of 
chemical bonds. Atoms bonded together represent a lower state of total energy than those same 
atoms existing separately, all other factors being equal. Thus, when separate atoms join together to 
form a molecule, they go from a high state of energy to a low state of energy, releasing the difference 
in energy in some form (heat, light, etc.). Conversely, an input of energy is required to break that 
chemical and force the atoms to separate. 

An example of this is the strong bond between two atoms of hydrogen (H) and one atom of oxygen 
(O), to form water (H2O). When hydrogen and oxygen atoms bond together to form water, they 
release energy. This, by definition, is an exothermic reaction, but we know it better as combustion: 
hydrogen is flammable in the presence of oxygen. 

A reversal of this reaction occurs when water is subjected to an electrical current, breaking water 
molecules up into hydrogen and oxygen gas molecules. This process of forced separation requires 
a substantial input of energy to accomplish, which by definition makes it an endothermic reaction. 
Specifically, the use of electricity to cause a chemical reaction is called electrolysis. 

Energy storage and release is the purpose of the so-called "hydrogen economy" where hydrogen 
is a medium of energy distribution. The reasoning behind a hydrogen economy is that different 
sources of energy will be used to separate hydrogen from oxygen in water, then that hydrogen will 
be transported to points of use and consumed as a fuel, releasing energy. All the energy released by 
the hydrogen at the point of use comes from the energy sources tapped to separate the hydrogen 
from oxygen in water. Thus, the purpose of hydrogen in a hydrogen economy is to function as an 
energy storage and transport medium. The fundamental principle at work here is the energy stored 
in chemical bonds: invested in the separation of hydrogen from oxygen, and later returned in the 
re-combination of hydrogen and oxygen back into water. 

The fact that hydrogen and oxygen as separate gases possess potential energy does not mean 
they are guaranteed to spontaneously combust when brought together. By analogy, just because 
rocks sitting on a hillside possess potential energy (by virtue of being elevated above the hill's base) 
does not means all rocks in the world spontaneously roll downhill. Some rocks need a push to get 
started because they are caught on a ledge or resting in a hole. Likewise, many exothermic reactions 
require an initial investment of energy before they can proceed. In the case of hydrogen and oxygen, 
what is generally needed is a spark to initiate the reaction. This initial requirement of input energy 
is called the activation energy of the reaction. 

Activation energy may be shown in graphical form. For an exothermic reaction, it appears as a 
"hill" that must be climbed before the total energy can fall to a lower (than original) level: 



Exothermic reaction 





For an endothermic reaction, activation energy is much greater, a part of which never returns 
but is stored in the reaction products as potential energy: 

Endothermic reaction 





A catalyst is a substance that works to minimize activation energy in a chemical reaction without 
being altered by the reaction itself. Catalysts are popularly used in industry to accelerate both 
exothermic and endothermic reactions, reducing the gross amount of energy that must be initially 
input to a process to make a reaction occur. A common example of a catalyst is the catalytic 



converter installed in the exhaust pipe of an automobile engine, helping to reduce oxidize unburnt 
fuel molecules and certain combustion products such as carbon monoxide (CO) to compounds which 
are not as polluting. Without a catalytic converter, the exhaust gas temperature is not hot enough 
to overcome the activation energy of these reactions, and so they will not occur (at least not at the 
rate necessary to make a significant difference). The presence of the catalyst allows the reactions to 
take place at standard exhaust temperatures. 

The effect of a catalyst on activation energy may be shown by the following graphs, the dashed- 
line curve showing the energy progression with a catalyst and the solid-line curve showing the 
reaction progressing without the benefit of a catalyst: 

Exothermic reaction 


Endothermic reaction 





2.6 Ions in liquid solutions 

Many liquid substances undergo a process whereby their constituent molecules split into positively 
and negatively charged ion pairs. Liquid ionic compounds split into ions completely or nearly 
completely, while only a small percentage of the molecules in a liquid covalent compound split into 
ions. The process of neutral molecules separating into ion pairs is called dissociation when it happens 
in ionic compounds, and ionization when it happens to covalent compounds. 

Molten salt (NaCl) is an example of the former, while pure water (H2O) is an example of the 
latter. The large presence of ions in molten salt explains why it is a good conductor of electricity, 
while the comparative lack of ions in pure water explains why it is often considered an insulator. In 
fact, the electrical conductivity of a liquid substance is the definitive test of whether it is an ionic 
or a covalent ("molecular") substance. 

Pure water ionizes into positive hydrogen ions 2 (H + ) and negative hydroxyl ions (OH - ). At 
room temperature, the concentration of hydrogen and hydroxyl ions in a sample of pure water is 
quite small: a molarity of 10 -7 M (moles per liter) each. 

Given the fact that pure water has a mass of 1 kilogram (1000 grams) per liter, and one mole 
of pure water has a mass of 18 grams, we must conclude that there are approximately 55.56 moles 
of water molecules in one liter (55.56 M). If only 10 -7 moles of those molecules ionize at room 
temperature, that represents an extremely small percentage of the total: 

10 -7 M 

— = 0.0000000018 = 0.00000018% = 0.0018 ppm (parts per million) 

It is not difficult to see why pure water is such a poor conductor of electricity. With so few 
ions available to act as charge carriers, the water is practically an insulator. The vast majority of 
water molecules remain un-ionized and therefore cannot transport electric charges from one point 
to another. 

The molarity of both hydrogen and hydroxyl ions in a pure water sample increases with increasing 
temperature. For example, at 60° C, the molarity of hydrogen and hydroxyl ions increases to 3.1 x 
10 -7 M, which is still only 0.0056 parts per million, but definitely larger than the concentration at 
room temperature (25° C). 

2 Actually, the more common form of positive ion in water is hydronium: H30 + , but we often simply refer to the 
positive half of an ionized water molecule as hydrogen (H + ). 

2.7. PH 69 

2.7 pH 

Hydrogen ion activity in aqueous (water-based) solutions is a very important parameter for a wide 
variety of industrial processes. Hydrogen ions are always measured on a logarithmic scale, and 
referred to as pH. 

Free hydrogen ions (H + ) are rare in a liquid solution, and are more often found attached to 
whole water molecules to form a positive ion called hydronium (HaO" 1 "). However, process control 
professionals usually refer to these positive ions simply as "hydrogen" even though the truth is a bit 
more complicated. 

pH is mathematically defined as the negative common logarithm of hydrogen ion activity in a 
solution. Hydrogen ion activity is expressed as a molarity (number of moles of active ions per liter 
of solution), with "pH" being the unit of measurement for the logarithmic result: 

pH = - log[H+] 

For example, an aqueous solution with an active hydrogen concentration of 0.00044 M has a pH 
value of 3.36 pH. 

Water is a covalent compound, and so there is little separation of water molecules in liquid form. 
Most of the water molecules remain as whole molecules (H2O) while a very small percentage ionize 
into positive hydrogen ions (H + ) and negative hydroxyl ions (OH - ). The mathematical product 
of hydrogen and hydroxyl ion molarity in water is known as the ionization constant {K w ), and its 
value varies with temperature. 

At 25 degrees Celsius (room temperature), the value of K w is 1.0 x 10~ 14 . Since each one of 
the water molecules that does ionize in this absolutely pure water sample separates into exactly one 
hydrogen ion (H + ) and one hydroxyl ion (OH - ), the molarities of hydrogen and hydroxyl ions must 
be equal to each other. The equality between hydrogen and hydroxyl ions in a pure water sample 
means that pure water is neutral, and that the molarity of hydrogen ions is equal to the square root 
of K,„ : 

[H+] = V^ = Vl.O x 10- 14 = 1.0 x 10~ 7 M 

Since we know pH is defined as the negative logarithm of hydrogen ion activity, and we can be 
assured all hydrogen ions present in the solution will be "active" since there are no other positive 
ions to interfere with them, the pH value for water at 25 degrees Celsius is: 

pH of pure water at 25°C = - log(1.0 x 10~ 7 M) = 7.0 pH 

As the temperature of a pure water sample changes, the ionization constant changes as well. 
Increasing temperature causes more of the water molecules to ionize, resulting in a larger K w value. 
The following table shows K w values for pure water at different temperatures: 




K w 

0° C 

1.139 x KT 15 

5° C 

1.846 x KT 15 

10° C 

2.920 x 10~ 15 

15° C 

4.505 x 10~ 15 

20° C 

6.809 x 10~ 15 

25° C 

1.008 x 10" 14 

30° C 

1.469 x 10~ 14 

35° C 

2.089 x 10~ 14 

40° C 

2.919 x 10~ 14 

45° C 

4.018 x 10~ 14 

50° C 

5.474 x 10~ 14 

55° C 

7.296 x 10~ 14 

60° C 

9.614 x 10" 14 

This means that while any pure water sample is neutral (an equal number of positive hydrogen 
ions and negative hydroxyl ions), the pH value does change with temperature, and is only equal to 
7.0 pH at one particular temperature: 25° C. Based on the K w values shown in the table, pure water 
will be 6.51 pH at 60° C and 7.47 pH at freezing. 

If we add an electrolyte to a sample of pure water, (at least some of) the molecules of that 
electrolyte will separate into positive and negative ions. If the positive ion of the electrolyte happens 
to be a hydrogen ion (H + ), we call that electrolyte an acid. If the negative ion of the electrolyte 
happens to be a hydroxyl ion (OH - ), we call that electrolyte a caustic, or alkaline, or base. Some 
common acidic and alkaline substances are listed here, showing their respective positive and negative 
ions in solution: 

Sulfuric acid is an acid (produces H + in solution) 
H 2 S0 4 -» 2H+ + S0 4 2 ~ 

Nitric acid is an acid (produces H + in solution) 
HN0 3 -> H+ + N0 3 - 

Hydrocyanic acid is an acid (produces H + in solution) 
HCN -> H+ + CN" 

Hydrofluoric acid is an acid (produces H + in solution) 
HF -> H+ + F~ 

Lithium hydroxide is a caustic (produces OH in solution) 
LiOH -> Li+ + OH" 

Potassium hydroxide is a caustic (produces OH in solution) 
KOH -> K+ + OH" 

2.7. PH 


Sodium hydroxide is a caustic (produces OH in solution) 
NaOH -> Na+ + OH - 

Calcium hydroxide is a caustic (produces OH - in solution) 
Ca(OH) 2 -> Ca 2+ + 20H - 

When an acid substance is added to water, some of the acid molecules dissociate into positive 
hydrogen ions (H + ) and negative ions (the type of negative ions depending on what type of acid 
it is). This increases the molarity of hydrogen ions (the number of moles of H + ions per liter of 
solution). The addition of hydrogen ions to the solution also decreases the molarity of hydroxyl 
ions (the number of moles of OH - ions per liter of solution) because some of the water's OH - ions 
combine with the acid's H + ions to form deionized water molecules (H2O). 

If an alkaline substance (otherwise known as a caustic, or a base) is added to water, some of 
the alkaline molecules dissociate into negative hydroxyl ions (OH - ) and positive ions (the type of 
positive ions depending on what type of alkaline it is). This increases the molarity of OH~ ions in 
the solution, as well as decreases the molarity of hydrogen ions (again, because some of the caustic's 
OH - ions combine with the water's H + ions to form deionized water molecules, H2O). 

The result of this complementary effect (increasing one type of water ion, decreasing the other) 
keeps the overall ionization constant relatively constant, at least for dilute solutions. In other words, 
the addition of an acid or a caustic may change [H + ], but it has little effect on K w . 

A simple way to envision this effect is to think of a laboratory balance scale, balancing the 
number of hydrogen ions in a solution against the number of hydroxyl ions in the same solution: 


When the solution is pure water, this imaginary scale is balanced (neutral), with [H + ] = [OH - ]. 
Adding an acid to the solution tips the scale one way, while adding a caustic to the solution tips it 
the other way 3 . 

3 It should be noted that the solution never becomes electrically imbalanced with the addition of an acid or caustic. 
It is merely the balance of hydrogen to hydroxyl ions we are referring to here. The net electrical charge for the 
solution should still be zero after the addition of an acid or caustic, because while the balance of hydrogen to hydroxyl 
ions does change, that electrical charge imbalance is made up by the other ions resulting from the addition of the 
electrolyte (anions for acids, cations for caustics). The end result is still one negative ion for every positive ion (equal 
and opposite charge numbers) in the solution no matter what substance (s) we dissolve into it. 


If an electrolyte has no effect on the hydrogen and hydroxyl ion activity of an aqueous solution, 
we call it a salt. The following is a list of some common salts, showing their respective ions in 

Potassium chloride is a salt (produces neither H + nor OH - nor 2 ~ in solution) 
KC1 -» K+ + Cl- 

Sodium chloride is a salt (produces neither H + nor OH~ nor 2 ~ in solution) 

NaCl -> Na+ + Cl" 

Zinc sulfate is a salt (produces neither H + nor OH~ nor 2 ~ in solution) 
ZnS0 4 -» Zn+ + S0 4 " 

The addition of a salt to an aqueous solution should have no effect on pH, because the ions 
created neither add to nor take away from the hydrogen ion activity 4 . 

When both an acid and caustic are added to an aqueous solution, their tendency is to neutralize 
one another, the hydrogen ions liberated by the acid combining (and canceling) with the hydroxyl 
ions liberated by the caustic. The result of a perfectly balanced mix of acid and caustic is deionized 
water (H2O) and a salt. Such neutralizations are exothermic, owing to the decreased energy states 
of the hydrogen and hydroxyl ions after combination. 


Giancoli, Douglas C, Physics for Scientists & Engineers, Third Edition, Prentice Hall, Upper Saddle 
River, New Jersey, 2000. 

Weast, Robert C; Astel, Melvin J.; and Beyer, William H., CRC Handbook of Chemistry and 
Physics, 64th Edition, CRC Press, Inc., Boca Raton, FL, 1984. 

Whitten, Kenneth W.; Gailey, Kenneth D.; and Davis, Raymond E., General Chemistry, Third 
Edition, Saunders College Publishing, Philadelphia, PA, 1988. 

4 Exceptions do exist for strong concentrations, where hydrogen ions may be present in solution yet unable to react 
because of being "crowded out" by other ions in the solution. 

Chapter 3 

DC electricity 




3.1 Electrical voltage 

Voltage is the amount of specific potential energy available between two points in an electric circuit. 
Potential energy is energy that is potentially available to do work. Looking at this from a classical 
physics perspective, potential energy is what we accumulate when we lift a weight above ground 
level, or when we compress a spring: 

Mass (m) 




Ground level 

Elastic force 




Height raised 




In either case, potential energy is calculated by the work done in exerting a force over a parallel 
distance. In the case of the weight, potential energy (E p ) is the simple product of weight (gravity g 
acting on the mass m) and height (h): 

E p = mgh 

For the spring, things are a bit more complex. The force exerted by the spring against the 
compressing motion increases with compression (F = kx, where k is the elastic constant of the 
spring). It does not remain steady as the force of weight does for the lifted mass. Therefore, the 
potential energy equation is nonlinear: 




Releasing the potential energy stored in these mechanical systems is as simple as dropping the 
mass, or letting go of the spring. The potential energy will return to the original condition (zero) 
when the objects are at rest in their original positions. If either the mass or the spring were attached 
to a machine to harness the return-motion, that stored potential energy could be used to do useful 

Potential energy may be similarly defined and quantified for any situation where we exert a force 
over a parallel distance, regardless of where that force or the motivating distance comes from. For 
instance, the static cling you experience when you pull a cotton sock out of a dryer is an example of 
a force. By pulling that sock away from another article of clothing, you are doing work, and storing 
potential energy in the tension between that sock and the rest of the clothing. In a similar manner, 
that stored energy could be released to do useful tasks if we placed the sock in some kind of machine 
that harnessed the return motion as the sock went back to its original place on the pile of laundry 
inside the dryer. 



If we make use of non-mechanical means to move electric charge from one location to another, the 
result is no different. Moving attracting charges apart from one another means doing work (a force 
exerted over a parallel distance) and storing potential energy in that physical tension. When we use 
chemical reactions to move electrons from one metal plate to another in a solution, or when we spin 
a generator and electro-magnetically motivate electrons to seek other locations, we impart potential 
energy to those electrons. We could express this potential energy in the same unit as we do for 
mechanical systems (the Joule). However, it is actually more useful to express the potential energy 
in an electric system in terms of how many joules are available per a specific quantity of electric 
charge (a certain number of electrons). This measure of specific potential energy is simply called 
electric potential or voltage, and we measure it in units of Volts, in honor of the Italian physicist 
Alessandro Volta, inventor of the first electrochemical battery. 

1 Volt 

1 Joule of potential energy 
1 Coulomb of electric charge 

In other words, if we forced 1 Coulomb's worth of electrons (6.24 x 10 18 of them, to be exact) 
away from a positively-charged place, and did one Joule's worth of work in the process, we would 
have generated one Volt of electric potential. 

Electric potential (voltage) and potential energy share a common, yet confusing property: both 
quantities are fundamentally relative between two physical locations. There is really no such thing 
as specifying a quantity of potential energy at a single location. The amount of potential energy 
in any system is always relative between two different points. If I lift a mass off the ground, I can 
specify its potential energy, but only in relation to its former position on the ground. The amount of 
energy that mass is potentially capable of releasing by free-fall depends on how far it could possibly 
fall. To illustrate, imagine lifting a 1 kilogram mass 1 meter off the ground. That 1-kilo mass weighs 
9.8 Newtons on Earth, and the distance lifted was 1 meter, so the potential energy stored in the 
mass is 9.8 joules, right? Consider the following scenario: 

Mass (m - 1 kg) 

Weight 1 

(mg = 9.8 Newtons) ^^ 

Height raised 
(h = 1 meter) 


300 meters to bottom 



0.5 meters 

7 / / / / 



If we drop the mass over the spot we first lifted it from, it will release all the potential energy 
we invested in it: 9.8 joules. But what if we carry it over to the table and release it there? Since 
now it can only fall half a meter, it will only release 4.9 joules in the process. How much potential 
energy did the mass have while suspended above that table? What if we carry it over to the edge of 
the cliff and release it there? Falling 301 meters, it will release 2.95 kilojoules (kj) of energy. How 
much potential energy did the mass have while suspended over the cliff? 

As you can see, potential energy is a relative quantity. We must know the mass's position relative 
to its falling point before we can quantify its potential energy. Likewise, we must know an electric 
charge's position relative to its return point before we can quantify the voltage it has. Consider a 
series of batteries connected as shown: 



The voltage as measured between any two points directly across a single battery will be 1.5 volts: 
Vab = 1.5 volts 
Vbc = 1-5 volts 
Vcd =1.5 volts 

If, however, we span more than one battery with our voltmeter connections, our voltmeter will 
register more than 1.5 volts: 
Vac = 3.0 volts 
Vbd = 3.0 volts 
Vad =4.5 volts 



There is no such thing as "voltage" at a single point in a circuit. The concept of voltage has 
meaning only between pairs of points in a circuit, just as the concept of potential energy for a mass 
has meaning only between two physical locations: where the mass is, and where it could potentially 
fall to. 

Things get interesting when we connect voltage sources in different configurations. Consider the 
following example, identical to the previous illustration except the middle battery has been reversed: 



Note the "+" and "-" signs next to the ends of the batteries. These signs show the polarity of 
each battery's voltage. Also note how the two voltmeter readings are different from before. Here we 
see an example of negative potential with the middle battery connected in opposition to the other 
two batteries. While the top and bottom batteries are both "lifting" electric charges to greater 
potential (going from point D to point A), the middle battery is decreasing potential from point C 
to point B. It's like taking a step forward, then a step back, then another step forward. Or, perhaps 
more appropriately, like lifting a mass 1.5 meters up, then setting it down 1.5 meters, then lifting 
it 1.5 meters up again. The first and last steps accumulate potential energy, while the middle step 
releases potential energy. 

This explains why it is important to install multiple batteries the same way into battery-powered 
devices such as radios and flashlights. The batteries' voltages are supposed to add to a make a larger 
total required by the device. If one or more batteries are placed backwards, potential will be lost 
instead of gained, and the device will not receive enough voltage. 

Here we must pay special attention to how we use our voltmeter, since polarity matters. All 



voltmeters are standardized with two colors for the test leads: red and black. To make sense of the 
voltmeter's indication, especially the positive or negative sign of the indication, we must understand 
what the red and black test lead colors mean: 

A positive reading indicates a gain 
in potential from black to red. 

A negative reading indicates a loss 
in potential from black to red. 


Connecting these test leads to different points in a circuit will tell you whether there is potential 
jain or potential loss from one point (black) to the other point (red). 


3.2 Electrical current 


Current is the name we give to the motion of electric charges from a point of high potential to a 
point of low potential. All we need to form an electric current is a source of potential (voltage) 
and some electric charges that are free to move between the poles of that potential. For instance, if 
we connected a battery to two metal plates, we would create an electric field between those plates, 
analogous to a gravitational field except that it only acts on electrically charged objects, while 
gravity acts on anything with mass. A free charge placed between those plates would "fall" toward 
one of the plates just as a mass would fall toward a larger mass: 

Gravitational field 



Metal plate 



Metal plate 

An electric charge will "fall" in an electric field 
just as a mass will fall in a gravitational field. 

Some substances, most notably metals, have very mobile electrons. That is, the outer (valence) 
electrons are very easily dislodged from the parent atoms to drift to and fro throughout the material. 
In fact, the electrons of metals are so free that physicists sometimes refer to the structure of a metal 
as atoms floating in a "sea of electrons" . The electrons are almost fluid in their mobility throughout 
a solid metal object, and this property of metals may be exploited to form definite pathways for 
electric currents. 

If the poles of a voltage source are joined by a continuous path of metal, the free electrons within 
that metal will drift toward the positive pole (electrons having a negative charge, opposite charges 
attracting one another): 



Direction of 
electron motion I 
inside metal ▼ 

If the source of this voltage is continually replenished by chemical energy, mechanical energy, or 
some other form of energy, the free electrons will continually loop around this circular path. We call 
this unbroken path an electric circuit. 

We typically measure the amount of current in a circuit by the unit of amperes, or amps for 
short (named in honor of the French physicist Andre Ampere. One ampere of current is equal to one 
coulomb of electric charge (6.24 x 10 18 electrons) moving past a point in a circuit for every second 
of time. 

Like masses falling toward a source of gravity, these electrons continually "fall" toward the 
positive pole of a voltage source. After arriving at that source, the energy imparted by that source 
"lifts" the electrons to a higher potential state where they once again "fall down" to the positive 
pole through the circuit. 

Like rising and falling masses in a gravitational field, these electrons act as carriers of energy 
within the electric field of the circuit. This is very useful, as we can use them to convey energy 
from one place to another, using metal wires as conduits for this energy. This is the basic idea 
behind electric power systems: a source of power (a generator) is turned by some mechanical engine 
(windmill, water turbine, steam engine, etc.), creating an electric potential. This potential is then 
used to motivate free electrons inside the metal wires to drift in a common direction. The electron 
drift is conveyed in a circuit through long wires, where they can do useful work at a load device such 
as an electric motor, light bulb, or heater. 

(Turned by an engine) 





, Current 





(Turns a conveyor belt 
or other mechanical load) 


Given the proper metal alloys, the friction that electrons experience within the metal wires may 
be made very small, allowing nearly all the energy to be expended at the load (motor) , with very little 
wasted along the path (wires) . This makes electricity the most efficient means of energy transport 

The electric currents common in electric power lines may range from hundreds to thousands of 
amperes. The currents conveyed through power receptacles in your home typically are no more 
than 15 or 20 amperes. The currents in the small battery-powered circuits you will build are even 
less: fractions of an ampere. For this reason, we commonly use the metric prefix milli (one one- 
thousandth) to express these small currents. For instance, 10 milliamperes is 0.010 amperes, and 
500 milliamperes is one-half of an ampere. 



3.2.1 Electron versus conventional flow 

When Benjamin Franklin advanced his single-fluid theory of electricity, he defined "positive" and 
"negative" as the surplus and deficiency of electric charge, respectively. These labels were largely 
arbitrary, as Mr. Franklin had no means of identifying the actual nature of electric charge carriers 
with the primitive test equipment and laboratory techniques of his day. As luck would have it, 
his hypothesis was precisely opposite of the truth for metallic conductors, where electrons are the 
dominant charge carrier. 

This means that in an electric circuit consisting of a battery and a light bulb, electrons slowly 
move from the negative side of the battery, through the metal wires, through the light bulb, and on 
to the positive side of the battery as such: 

Direction of electron flow 

Unfortunately, scientists and engineers had grown accustomed to Franklin's false hypothesis long 
before the true nature of electric current in metallic conductors was discovered. Their preferred 
notation was to show electric current flowing from the positive pole of a source, through the load, 
returning to the negative pole of the source: 


Direction of conventional flow 

This relationship between voltage polarity marks and conventional flow current makes more 
intuitive sense than electron flow notation, because it is reminiscent of fluid pressure and flow 




Conventional flow current notation 

Voltage _ 

(J) Light 

Conventional flow current notation 



I — > Fluid motion 



Fluid motion 

inn ^ 

If we take the "+" sign to represent more pressure and the "-" sign to represent less pressure, 
it makes perfect sense that fluid should move from the high-pressure (discharge) port of the pump 
through the hydraulic "circuit" and back to the low-pressure (suction) port of the pump. It also 
makes perfect sense that the upstream side of the valve (a fluid restriction) will have a greater 
pressure than the downstream side of the valve. In other words, conventional flow notation best 
honors Mr. Franklin's original intent of modeling current as though it were a fluid, even though he 
was later proven to be mistaken in the case of metallic conductors where electrons are the dominant 
charge carrier. 

This convention was so well-established in the electrical engineering realm that it held sway 
despite the discovery of electrons. Engineers, who create the symbols used to represent the electronic 
devices they invent, consistently chose to draw arrows in the direction of conventional flow rather 
than electron flow. In each of the following symbols, the arrow heads point in the direction that 
positive charge carriers would move (opposite the direction that electrons actually move): 



NPN bipolar 


PNP bipolar 

* 4 







This stands in contrast to electronics technicians, who historically have been taught using electron 
flow notation. I remember sitting in a technical school classroom being told by my teacher to always 
imagine the electrons moving against the arrows of the devices, and wondering why it mattered. 

It is truly a sad situation when the members of two branches within the same field do not agree 
on something as fundamental as the convention used to denote flow in diagrams. It is even worse 
when people within the field argue over which convention is best. So long as one is consistent with 
their convention and with their thinking, it does not matter! Many fine technologists may be found 
on either side of this "fence," and some are adept enough to switch between both without getting 

For what it's worth, I personally prefer conventional flow notation. The only objective arguments 
I have in favor of this preference are as follows: 

• Conventional flow notation makes more intuitive sense to someone familiar with fluid systems 
(as all instrument technicians need to be!). 

• Conventional flow notation matches all device arrows; no need to "go against the arrow" when 
tracing current in a schematic diagram. 

• Conventional flow notation is consistent with the "right-hand rule" for vector cross products 
(which are essential for understanding electromagnetics at advanced academic levels). The 
so-called "left-hand rule" taught to students learning electron flow notation is mathematically 
wrong, and must be un-learned if the student ever progresses to the engineering level in his or 
her studies. 

• Conventional flow notation is the standard for modern manufacturers' 
(reference manuals, troubleshooting guides, datasheets, etc.) 1 . 


1 l have yet to read a document of any kind written by an equipment manufacturer that uses electron flow notation, 
and this is after scrutinizing literally hundreds of documents looking for this exact detail! For the record, though, most 
technical documents do not bother to draw a direction for current at all, leaving it to the imagination of the reader 
instead. It is only when a direction must be drawn that one sees a strong preference in industry for conventional flow 


N r > 

• Conventional flow notation makes sense of the descriptive terms sourcing and sinking. 

This last point merits further investigation. The terms "sourcing" and "sinking" are often used 
in the study of digital electronics to describe the direction of current in a switching circuit. A circuit 
that "sources" current to a load is one where the direction of conventional flow points outward from 
the sourcing circuit to the load device. 

For example, here are two schematic diagrams showing two different kinds of electronic proximity 
switch. The first switch sinks current in from the LED through its output terminal, through its 
transistor, and down to ground. The second switch sources current from the positive supply terminal 
through its transistor and out to the LED through its output terminal (note the direction of the 
thick arrow near the output screw terminal in each circuit): 

"Sinking" output 
proximity switch 




Current "sinks" down to 
ground through the switch 






"Sourcing" output Switch "sources" current 

proximity switch out to the load device 


Output L ^p (l) 24VDC 

®ww (B) 


These terms simply make no sense when viewed from the perspective of electron flow notation. 
If you were to actually trace the directions of the electrons, you would find that a device "sourcing" 
current has electrons flowing into its connection terminal, while a device "sinking" current sends 
electrons out to another device where they travel (up) to a point of more positive potential. 


In fact, the association between conventional flow notation and sourcing/sinking descriptions is 
so firm that I have yet to see a professionally published textbook on digital circuits that uses electron 
flow 2 . This is true even for textbooks written for technicians and not engineers! 

Once again, though, it should be understood that either convention of current notation is 
adequate for circuit analysis. I dearly wish this horrible state of affairs would come to an end, 
but the plain fact is that it will not. Electron flow notation may have the advantage of greater 
correspondence to the actual state of affairs (in the vast majority of circuits), but conventional flow 
has the weight of over a hundred years of precedent, cultural inertia, and convenience. No matter 
which way you choose to think, at some point you will be faced with the opposing view. 

Pick the notation you like best, and may you live long and prosper. 

2 If by chance I have missed anyone's digital textbook that does use electron flow, please accept my apologies, 
can only speak of what I have seen myself. 


3.3 Electrical resistance and Ohm's Law 

To review, voltage is the measure of potential energy available to electric charges. Current is the 
uniform drifting of electric charges in response to a voltage. We can have a voltage without having 
a current, but we cannot have a current without first having a voltage to motivate it 3 . Current 
without voltage would be equivalent to motion without a motivating force. 

When electric charges move through a material such as metal, they will naturally encounter some 
friction, just as fluid moving through a pipe will inevitably encounter friction 4 . We have a name for 
this friction to electrical charge motion: resistance. Like voltage and current, resistance has its own 
special unit of measurement: the ohm, named in honor of the German physicist Georg Simon Ohm. 

At this point it would be good to summarize and compare the symbols and units we use for 
voltage, current, and resistance: 


Algebraic symbol 


Unit abbreviation 


V (or E) 





Ampere (or Amp) 






Ohm defined resistance as the mathematical ratio between applied voltage and resulting current: 

Verbally expressed, resistance is how much voltage it takes to force a certain rate of current 
through a conductive material. Many materials have relatively stable resistances, while others do 
not. Devices called resistors are sold which are manufactured to possess a very precise amount of 
resistance, for the purpose of limiting current in circuits (among other things). 

Here is an example of Ohm's Law in action: calculate the amount of current in a circuit with a 
voltage source of 25 V and a total resistance of 3500 Q. Taking 25 volts and dividing by 3500 ohms, 
you should arrive at a result of 0.007143 amperes, or 7.143 milliamperes (7.143 mA). 

One of the most challenging aspect of Ohm's Law is remembering to keep all variables in context. 
This is a common problem for many students when studying physics as well: none of the equations 
learned in a physics class will yield the correct results unless all the variables relate to the same 
object or situation. For instance, it would make no sense to try to calculate the kinetic energy of a 
moving object (E = ^mv 2 ) by taking the mass of one object (m) and multiplying it by the square of 
the velocity of some other object (v 2 ). Likewise, with Ohm's Law, we must make sure the voltage, 
current, and resistance values we are using all relate to the same portion of the same circuit. 

If the circuit in question has only one source of voltage, one resistance, and one path for current, 
there cannot be any mix-ups. Expressing the previous example in a schematic diagram: 

3 Except in the noteworthy case of superconductivity, a phenomenon occurring at extremely low temperatures. 

4 Except in the noteworthy case of superfluidity, another phenomenon occurring at extremely low temperatures. 



7.143 mA <- 

25 v Ovoltage 




-> 7.143 mA 

3500 D. 

Note: arrows point in the direction of electron motion 

However, if we look at a more complex circuit, we encounter the potential for mix-ups: 

25 V 

3500 a 

1500 Q. 

Which resistance do we use to calculate current in this circuit? Do we divide our 25 volts by 
3500 ohms like we did last time, or do we divide it by 1500 ohms, or something entirely different? 
The answer to this question lies in the identification of voltages and currents. We know that the 25 
volt potential will be impressed across the total of the two resistances i?i and i?2, and since there is 
only one path for current they must share the same current. Thus, we actually have three voltages 
(Vi, V2, and Vtotai), three resistances {R\, R2, and Rtotai), and only one current (J): 



25 V ( - ) V 




V, R, > 3500 ft 

V, R 2 S 1500 ft 

Note: arrows point in the direction of electron motion 

Manipulating the Ohm's Law equation originally given (R = y) to solve for V, we end up with 
three equations for this circuit: 

Vtotal = IRtotal = I(Rl + R2) 

V x = IR, 

V 2 = IR 2 

Thus, the current in this circuit is 5 milliamps (5 mA), the voltage across resistor R\ is 17.5 
volts, and the voltage across resistor R 2 is 7.5 volts. 



3.4 Series versus parallel circuits 

In addition to Ohm's Law, we have a whole set of rules describing how voltages, currents, and 
resistances relate in circuits comprised of multiple resistors. These rules fall evenly into two 
categories: series circuits and parallel circuits. The two circuit types are shown here, with squares 
representing any type of two-terminal electrical component: 

Series circuit 



Parallel circuit 

Equipotential points 

-f f f f- 

Equipotential points 

The defining characteristic of a series electrical circuit is that it has just one path for current. 
This means there can be only one value for current anywhere in the circuit, the exact same current 
for all components at any given time 5 . The principle of current being the same everywhere in a 
series circuit is actually an expression of a more fundamental law of physics: the Conservation of 
Charge, which states that electric charge cannot be created or destroyed. In order for current to 
have different values at different points in a series circuit indefinitely, electric charge would have to 
somehow appear and disappear to account for greater rates of charge flow in some areas than in 
others. It would be the equivalent of having different rates of water flow at different locations along 
one length of pipe 6 . 

Series circuits are defined by having only one path for current, and this means the steady-state 
current in a series circuit must be the same at all points of that circuit. It also means that the sum 
of all voltages dropped by load devices must equal the sum total of all source voltages, and that the 
total resistance of the circuit will be the sum of all individual resistances: 

5 Interesting exceptions do exist to this rule, but only on very short time scales, such as in cases where we examine 
the a transient (pulse) signal nanosecond by nanosecond, and/or when very high-frequency AC signals exist over 
comparatively long conductor lengths. 

6 Those exceptional cases mentioned earlier in the footnote are possible only because electric charge may be 
temporarily stored and released by a property called capacitance. Even then, the law of charge conservation is 
not violated because the stored charges re-emerge as current at later times. This is analogous to pouring water into 
a bucket: just because water is poured into a bucket but no water leaves the bucket does not mean that water is 
magically disappearing! It is merely being stored, and can re-emerge at a later time. 



Series circuit (resistors connected in-line) 



v 2 


R 2 : 

v 3 

r 3 ; 

v 4 

R 4 : 

— >- 

Voltages add up to equal the total 

v total = v 1 + v 2 + ... + v n 

Current is the same throughout 

Itotal - M - ^2 - ■ • • = In 

Resistances add up to equal the total 
Rtotai - R i + R 2 + • • • + R n 

The defining characteristic of a parallel circuit, by contrast, is that all components share the 
same two equipotential points. "Equipotential" simply means "at the same potential" which points 
along an uninterrupted conductor must be 7 . This means there can be only one value of voltage 
anywhere in the circuit, the exact same voltage for all components at any given time 8 . The principle 
of voltage being the same across all parallel-connected components is (also) an expression of a more 
fundamental law of physics: the Conservation of Energy, in this case the conservation of specific 
potential energy which is the definition of voltage. In order for voltage to differ between parallel- 
connected components, the potential energy of charge carriers would have to somehow appear and 
disappear to account for lesser and greater voltages. It would be the equivalent of having a "high 
spots" and "low spots" of water mysteriously appear on the quiet surface of a lake, which we know 
cannot happen because water has the freedom to move, meaning any high spots would rush to fill 
any low spots 9 . 

The sum of all component currents must equal the total current in a parallel circuit, and total 
resistance will be less than the smallest individual resistance value: 

7 An ideal conductor has no resistance, and so there is no reason for a difference of potential to exist along a 
pathway where nothing stands in the way of charge motion. If ever a potential difference developed, charge carriers 
within the conductor would simply move to new locations and neutralize the potential. 

8 Again, interesting exceptions do exist to this rule on very short time scales, such as in cases where we examine 
the a transient (pulse) signal nanosecond by nanosecond, and/or when very high-frequency AC signals exist over 
comparatively long conductor lengths. 

9 The exceptional cases mentioned in the previous footnote exist only because the electrical property of inductance 
allows potential energy to be stored in a magnetic field, manifesting as a voltage different along the length of a 
conductor. Even then, the law of energy conservation is not violated because the stored energy re-emerges at a later 



Parallel circuit (resistors connected across each other) 




R 2 

h > 

R 3 




Voltage is the same throughout 
V total = V 1 = V 2 = ... = V n 

Currents add up to equal the total 

Itotal - II + I2 + • • • + In 

Resistances diminish to equal the total 
R total = ( R i + R 2 + • ■ ■ + R n ) 

The rule for calculating total resistance in a parallel circuit perplexes many students with its 
weird compound reciprocal notation. There is a more intuitive way to understand this rule, and it 
involves a different quantity called conductance, symbolized by the letter G. 

Conductance is defined as the reciprocal of resistance; that is, a measure of how easily electrical 
charge carriers may move through a substance. If the electrical resistance of an object doubles, then 
it now has half the conductance it did before: 

It should be intuitively apparent that conductances add in parallel circuits. That is, the total 
amount of conductance for a parallel circuit must be the sum total of all individual conductances, 
because the addition of more conductive pathways must make it easier overall for charge carriers to 
move through the circuit. Thus, 



G\ + Gi + ■ • • + G n 

The formula shown here should be familiar to you. It has the same form as the total resistance 
formula for series circuits. Just as resistances add in series (more series resistance makes the overall 
resistance to current increase), conductances add in parallel (more conductive branches makes the 
overall conductance increase). 

Knowing that resistance is the reciprocal of conductance, we may substitute -i for G wherever 
we see it in the conductance equation: 


Rtotal Ri R2 R 

Now, to solve for Rtotal, we need to reciprocate both sides: 

1 1 

Ri + Ra 




total 11 i 

— — I — - — h • • • H — — 

For both series and parallel circuits, total power dissipated by all load devices is equal to the 
total power delivered by all source devices. The configuration of a circuit is irrelevant to the balance 
between power supplied and power lost, because this balance is an expression of the Law of Energy 



3.5 Kirchhoff 's Laws 

Two extremely important principles in electric circuits were codified by Gustav Robert Kirchhoff in 
the year 1847, known as Kirchhoff 's Laws. His two laws refer to voltages and currents in electric 
circuits, respectively. 

Kirchhoff's Voltage Law states that the algebraic sum of all voltages in a closed loop is equal to 
zero. Another way to state this law is to say that for every rise in potential there must be an equal 
fall, if we begin at any point in a circuit and travel in a loop back to that same starting point. 

An analogy for visualizing Kirchhoff's Voltage Law is hiking up a mountain. Suppose we start 
at the base of a mountain and hike to an altitude of 5,000 feet to set up camp for an overnight stay. 
Then, the next day we set off from camp and hike further up another 3,500 feet. Deciding we've 
climbed high enough for two days, we set up camp again and stay the night. The next day we hike 
down 6,200 feet to a third location and camp once gain. On the fourth day we hike back to our 
original starting point at the base of the mountain. We can summarize our hiking adventure as a 
series of rises and falls like this: 



Altitude gain/loss 

Day 1 


+5,000 feet 

Day 2 


+3,500 feet 

Day 3 


-6,200 feet 

Day 4 


-2,300 feet 




Of course, no one would brag to their friends that they spent four days hiking a total altitude 
of feet, so people generally speak in terms of the highest point reached: in this case 8,500 feet. 
However, if we track each day's gain or loss in algebraic terms (maintaining the mathematical sign, 
either positive or negative), we see that the end sum is zero (and indeed must always be zero) if we 
finish at our starting point. 

If we view this scenario from the perspective of potential energy as we lift a constant mass from 
point to point, we would conclude that we were doing work on that mass (i.e. investing energy in 
it by lifting it higher) on days 1 and 2, but letting the mass do work on us (i.e. releasing energy by 



lowering it) on days 3 and 4. After the four-day hike, the net potential energy imparted to the mass 
is zero, because it ends up at the exact same altitude it started at. 

Let's apply this principle to a real circuit, where total current and all voltage drops have already 
been calculated for us: 

4 mA 







1.5 k£l 


2 V 


6 V 

Arrow shows current in the direction 
of conventional flow notation 

If we trace a path ABCDEA, we see that the algebraic voltage sum in this loop is zero: 


Voltage gain/loss 


- 4 volts 


- 6 volts 


+ 5 volts 


- 2 volts 


+ 7 volts 



We can even trace a path that does not follow the circuit conductors or include all components, 
such as EDCBE, and we will see that the algebraic sum of all voltages is still zero: 


Voltage gain/loss 


+ 2 volts 


- 5 volts 


+ 6 volts 


- 3 volts 



Kirchhoff's Voltage Law is often a difficult subject for students, precisely because voltage itself 
is a difficult concept to grasp. Remember that there is no such thing as voltage at a single point; 
rather, voltage exists only as a differential quantity. To intelligently speak of voltage, we must refer 
to either a loss or gain of potential between two points. 

Our analogy of altitude on a mountain is particularly apt. We cannot intelligently speak of some 
point on the mountain as having a specific altitude unless we assume a point of reference to measure 
from. If we say the mountain summit is 9,200 feet high, we usually mean 9,200 feet higher than sea 
level, with the level of the sea being our common reference point. However, our hiking adventure 



where we climbed 8,500 feet in two days did not imply that we climbed to an absolute altitude of 
8,500 feet above sea level. Since I never specified the sea-level altitude at the base of the mountain, 
it is impossible to calculate our absolute altitude at the end of day 2. All you can tell from the 
data given is that we climbed 8,500 feet above the mountain base, wherever that happens to be with 
reference to sea level. 

So it is with electrical voltage as well: most circuits have a point labeled as ground where all 
other voltages are referenced. In DC-powered circuits, this ground point is often the negative pole of 
the DC power source 10 . Voltage is fundamentally a quantity relative between two points: a measure 
of how much potential has increased or decreased moving from one point to another. 

Kirchhoff's Current Law is a much easier concept to grasp. This law states that the algebraic 
sum of all currents at a junction point (called a node) is equal to zero. Another way to state this 
law is to say that for every electron entering a node, one must exit somewhere. 

An analogy for visualizing Kirchhoff's Current Law is water flowing into and out of a "tee" 

300 GPM 

230 GPM 


70 GPM 

So long as there are no leaks in this piping system, every drop of water entering the tee must 
be balanced by a drop exiting the tee. For there to be a continuous mis-match between flow rates 
would imply a violation of the Law of Mass Conservation. 

Let's apply this principle to a real circuit, where all currents have been calculated for us: 

10 But not always! There do exist positive-ground systems, particularly in telephone circuits and in some early 
automobile electrical systems. 



I Ml 


4 mA 

4 mA D 

Arrows show currents in the direction 
of conventional flow notation 

At nodes where just two wires connect (such as points A, B, and C), the amount of current going 
in to the node exactly equals the amount of current going out (4 mA, in each case). At nodes where 
three wires join (such as points D and E), we see one large current and two smaller currents (one 
4 mA current versus two 2 mA currents) , with the directions such that the sum of the two smaller 
currents form the larger current. 


3.6 Electrical sources and loads 

By definition, and source is a device that inputs energy into a system, while a load is a device that 
extracts energy from a system. Examples of typical electrical sources include generators, photovoltaic 
cells, thermopiles, and primary-cell batteries. Examples of typical electrical loads include resistors, 
lamps, and electric motors. 

In a working circuit, electrical sources and loads may be easily distinguished by comparison of 
their current directions and voltage drop polarities. An electrical source always manifests a voltage 
polarity in a direction that assists the direction of charge flow. An electrical source always manifests 
a voltage polarity in a direction that opposes the direction of charge flow. 

The convention used to designate direction of current (charge flow) becomes very important here. 
Since there are two commonly accepted notations - electron flow and "conventional" flow, exactly 
opposite of each other - it is easy to become confused. 

First we see a diagram showing a source and a load, using electron flow notation. Electrons, 
being negatively charged particles, are repelled by the negative (-) poles of both source and load, 
and attracted to the positive (+) poles of both source and load. The difference between source and 
load is that the source device motivates the flow of electrons while the load device resists the flow 
of electrons: 

Shown using electron flow notation 

Source + J + Load 

Generator (z) S Resistor 

Electrons are repelled by the (-) poles 
and attracted to the (+) poles 

Next we see a diagram showing the same source and load, this time using "conventional" flow 
notation to designate the direction of current. Here we must imagine positively-charged carriers 
moving through the wires instead of electrons. These positive charge carriers are repelled by any 
positive (+) pole and attracted to any negative (-) pole. Viewed in this light, we see the exact same 
principle at work: the source device is seen to motivate the flow of these positive charge carriers 
while the load device resists the flow: 



Shown using conventional flow notation 


Generator (_ 






Positive charge carriers are repelled by the 
(+) poles and attracted to the (-) poles 

In later sections, we encounter devices with the ability to act as sources and loads at different 
times. Both capacitors (see section 3.9 starting on page 108) and inductors (see section 3.10 starting 
on page 110) have the ability to temporarily contribute to and extract energy from electrical circuits, 
both having the ability to act as energy storage devices. 

3.7 Resistors 

Resistance is dissipative opposition to the flow of charge carriers. All conductors (except 
superconductors) possess some electrical resistance. The relationship between voltage, current, and 
resistance is known as Ohm's Law: 

V = IR 

Conductance (G) is the reciprocal of resistance: 



Resistors are devices expressly designed and manufactured to possess electrical resistance. They 
are constructed of a partially conductive material such as carbon or metal alloy. Resistors have power 
dissipation ratings as well as resistance ratings. Here are some schematic symbols for resistors: 

The amount of power dissipated by a resistance may be calculated as a function of either voltage 
or current, and is known as Joule's Law. 

P = IV 



I 2 R 



3.8 Bridge circuits 

A bridge circuit is basically a pair of voltage dividers where the circuit output is taken as the 
difference in potential between the two dividers. Bridge circuits may be drawn in schematic form in 
an H-shape or in a diamond shape, although the diamond configuration is more common: 

v . . r) 

v excitation \—y 






The voltage source powering the bridge circuit is called the excitation source. This source may 
be DC or AC depending on the application of the bridge circuit. The components comprising the 
bridge need not be resistors, either: capacitors, inductors, lengths of wire, sensing elements, and 
other component forms are possible, depending on the application. 

Two major applications exist for bridge circuits, which will be explained in the following 



3.8.1 Component measurement 

Bridge circuits may be used to test components. In this capacity, one of the "arms" of the bridge 
circuit is comprised of the component under test, while at least one of the other "arms" is made 
adjustable. The common Wheatstone bridge circuit for resistance measurement is shown here: 




Fixed resistors i?i and Ri are of precisely known value and high precision. Variable resistor 
R a djust has a labeled knob allowing for a person to adjust and read its value to a high degree of 
precision. When the ratio of the variable resistance to the specimen resistance equals the ratio of 
the two fixed resistors, the sensitive galvanometer will register exactly zero volts regardless of the 
excitation source's value. This is called a balanced condition for the bridge circuit: 






When the two resistance ratios are equal, the voltage drops across the respective resistances will 
also be equal. Kirchhoff 's Voltage Law declares that the voltage differential between two equal and 
opposite voltage drops must be zero, accounting for the meter's indication of balance. 

It would not be inappropriate to relate this to the operation of a laboratory balance-beam scale, 
comparing a specimen of unknown mass against a set of known masses. In either case, the instrument 
is merely comparing an unknown quantity against an (adjustable) known quantity, indicating a 
condition of equality between the two: 


Many legacy instruments were designed around the concept of a self-balancing bridge circuit, 
where an electric servo motor drove a potentiometer to achieve a balanced condition against the 
voltage produced by some process sensor. Analog electronic paper chart recorders often used this 
principle. Almost all pneumatic process instruments use this principle to translate the force of a 
sensing element into a variable air pressure. 

Modern bridge circuits are mostly used in laboratories for extremely precise component 
measurements. Very rarely will you encounter a Wheatstone bridge circuit used in the process 



3.8.2 Sensor signal conditioning 

A different application for bridge circuits is to convert the output of an electrical sensor into a 
voltage signal representing some physical measurement. This is by far the most popular use of 
bridge measurement circuits in industry, and here we see the same circuit used in an entirely different 
manner from that of the balanced Wheatstone bridge circuit. 

y ■ ■ r) 

v excitation \—y 

Here, the bridge will be balanced only when R sensor is at one particular resistance value. Unlike 
the Wheatstone bridge, which serves to measure a component's value when the circuit is balanced, 
this bridge circuit will probably spend most of its life in an unbalanced condition. The output 
voltage changes as a function of sensor resistance, which makes that voltage a reflection of the 
sensor's physical condition. In the above circuit, we see that the output voltage increases (positive 
on the top wire, negative on the bottom wire) as the resistance of R ae nsor increases. 

One of the most common applications for this kind of bridge circuit is in strain measurement, 
where the mechanical strain of an object is converted into an electrical signal. The sensor used here 
is a device known as a strain gauge: a folded wire designed to stretch and compress with the object 
under test, altering its electrical resistance accordingly. 



Test specimen 

When the specimen is stretched along its long axis, the metal wires in the strain gauge stretch 
with it, increasing their length and decreasing their cross-sectional area, both of which work to 
increase the wire's electrical resistance. This stretching is microscopic in scale, but the resistance 
change is measurable and repeatable within the specimen's elastic limit. In the above circuit example, 
stretching the specimen will cause the voltmeter to read upscale (as defined by the polarity marks). 
Compressing the specimen along its long axis has the opposite effect, decreasing the strain gauge 
resistance and driving the meter downscale. 

Strain gauges are used to precisely measure the strain (stretching or compressing motion) 
of mechanical elements. One application for strain gauges is the measurement of strain on 
machinery components, such as the frame components of an automobile or airplane undergoing 
design development testing. Another application is in the measurement of force in a device called 
a load cell. A "load cell" is comprised of one or more strain gauges bonded to the surface of a 
metal structure having precisely known elastic properties. This metal structure will stretch and 
compress very precisely with applied force, as though it were an extremely stiff spring. The strain 
gauges bonded to this structure measure the strain, translating applied force into electrical resistance 

You can see what a load cell looks like in the following photograph: 



Strain gauges are not the only dynamic element applicable to bridge circuits. In fact, any 
resistance-based sensor may be used in a bridge circuit to translate a physical measurement into 
an electrical (voltage) signal. Thermistors (changing resistance with temperature) and photocells 
(changing resistance with light exposure) are just two alternatives to strain gauges. 

It should be noted that the amount of voltage output by this bridge circuit depends both on the 
amount of resistance change of the sensor and the value of the excitation source. This dependency 
on source voltage value is a major difference between a sensing bridge circuit and a Wheatstone 
(balanced) bridge circuit. In a perfectly balanced bridge, the excitation voltage is irrelevant: the 
output voltage is zero no matter what source voltage value you use. In an unbalanced bridge circuit, 
however, source voltage value matters! For this reason, these bridge circuits are often rated in terms 
of how many millivolts of output they produce per volt of excitation per unit of physical measurement 
(microns of strain, newtons of stress, etc.). 

An interesting feature of a sensing bridge circuit is its ability to cancel out unwanted variables. 
In the case of a strain gauge, for example, mechanical strain is not the only variable affecting gauge 
resistance. Temperature also affects gauge resistance. Since we do not wish our strain gauge to also 
act as a thermometer (which would make measurements very uncertain - how would we differentiate 
the effects of changing temperature from the effects of changing strain?), we must find some way to 
nullify resistance changes due solely to temperature, such that our bridge circuit will respond only 
to changes in strain. The solution is to creatively use a "dummy" strain gauge as another arm of 
the bridge: 



The "dummy" gauge is attached to the specimen in such a way that it maintains the same 
temperature as the active strain gauge, yet experiences no strain. Thus, any difference in gauge 
resistances must be due solely to specimen strain. The differential nature of the bridge circuit 
naturally translates the differential resistance of the two gauges into one voltage signal representing 

If thermistors are used instead of strain gauges, this circuit becomes a differential temperature 
sensor. Differential temperature sensing circuits are used in solar heating control systems, to detect 
when the solar collector is hotter than the room or heat storage mass being heated. 

Sensing bridge circuits may have more than one active "arm" as well. The examples you have 
seen so far in this section have all been quarter-active bridge circuits. It is possible, however, to 
incorporate more than one sensor into the same bridge circuit. So long as the sensors' resistance 
changes are coordinated, their combined effect will be to increase the sensitivity (and often the 
linearity as well) of the measurement. 

For example, full-active bridge circuits are sometimes built out of four strain gauges, where each 
strain gauge comprises one arm of the bridge. Two of the strain gauges must compress and the 
other two must stretch under the application of the same mechanical force, in order that the bridge 
will become unbalanced with strain: 






Gauge 1 Tension 
Test specimen 


Gauge 3 


Gauge 1 

Gauge 3 

Not only does a full-active bridge circuit provide greater sensitivity and linearity than a quarter- 
active bridge, but it also naturally provides temperature compensation without the need for 
"dummy" strain gauges, since the resistances of all four strain gauges will change by the same 
proportion if the specimen temperature changes. 


3.9 Capacitors 

Any two electrical conductors separated by an insulating medium possess the characteristic called 
capacitance: the ability to store energy in the form of an electric field. Capacitance is symbolized 
by the capital letter C and is measured in the unit of the Farad (F). The relationship between 
capacitance, stored electric charge (Q), and voltage (V) is as follows: 

Q = CV 

For example, a capacitance having a value of 33 microfarads charged to a voltage of 5 volts would 
store an electric charge of 165 microcoulombs. 

Capacitance is a non-dissipative quantity. Unlike resistance, a pure capacitance does not dissipate 
energy in the form of heat; rather, it stores and releases energy from and to the rest of the circuit. 

Capacitors are devices expressly designed and manufactured to possess capacitance. They are 
constructed of a "sandwich" of conductive plates separated by an insulating dielectric. Capacitors 
have voltage ratings as well as capacitance ratings. Here are some schematic symbols for capacitors: 

Nonpolarized Polarized 


1 =b 

T T 

A capacitor's capacitance is related to the electric permittivity of the dielectric material 
(symbolized by the Greek letter "epsilon," e), the cross-sectional area of the overlapping plates 
(A), and the distance separating the plates (d): 


Capacitance adds when capacitors are connected in parallel. It diminishes when capacitors are 
connected in series: 

^parallel — ^1 ~T ^2 ~T ^n ^series — ~^[ ' J ' ' Y~ 

The relationship between voltage and current for a capacitor is as follows: 

i = c d ^ 


As such, capacitors oppose changes in voltage over time by creating a current. This behavior 
makes capacitors useful for stabilizing voltage in DC circuits. One way to think of a capacitor 

3.9. CAPACITORS 109 

in a DC circuit is as a temporary voltage source, always "wanting" to maintain voltage across its 
terminals at the same value. 

The amount of potential energy (E p , in units of joules) stored by a capacitor is proportional to 
the square of the voltage: 

E v = -CV 2 
p 2 

In an AC circuit, the amount of capacitive reactance (Xc) offered by a capacitor is inversely 
proportional to both capacitance and frequency: 

Xc= 2^fC 


3.10 Inductors 

Any conductor possesses a characteristic called inductance: the ability to store energy in the form 
of a magnetic field. Inductance is symbolized by the capital letter L and is measured in the unit of 
the Henry (H). 

Inductance is a non-dissipative quantity. Unlike resistance, a pure inductance does not dissipate 
energy in the form of heat; rather, it stores and releases energy from and to the rest of the circuit. 

Inductors are devices expressly designed and manufactured to possess inductance. They are 
typically constructed of a wire coil wound around a ferromagnetic core material. Inductors have 
current ratings as well as inductance ratings. Due to the effect of magnetic saturation, inductance 
tends to decrease as current approaches the rated maximum value in an iron-core inductor. Here 
are some schematic symbols for inductors: 

An inductor's inductance is related to the magnetic permeability of the core material (/i), the 
number of turns in the wire coil (AT), the cross-sectional area of the coil (A), and the length of the 
coil (I): 

_ nN 2 A 

Inductance adds when inductors are connected in series. It diminishes when inductors are 
connected in parallel: 

^series — -^1 "T -^2 T L-t n ^parallel — ~ J ~ 

The relationship between voltage and current for an inductor is as follows: 


As such, inductors oppose changes in current over time by dropping a voltage. This behavior 
makes inductors useful for stabilizing current in DC circuits. One way to think of an inductor in a 
DC circuit is as a temporary current source, always "wanting" to maintain current through its coil 
at the same value. 

The amount of potential energy (E p , in units of joules) stored by an inductor is proportional to 
the square of the current : 

E v = -LI 2 

p 2 

In an AC circuit, the amount of inductive reactance (Xr,) offered by an inductor is directly 
proportional to both inductance and frequency: 

X L = 2nfL 

3.10. INDUCTORS 111 


Boylestad, Robert L., Introductory Circuit Analysis, 9th Edition, Prentice Hall, Upper Saddle River, 
New Jersey, 2000. 


Chapter 4 

AC electricity 




4.1 RMS quantities 

It is often useful to be able to express the amplitude of an AC quantity such as voltage or current in 
terms that are equivalent to direct current (DC). Doing so provides an "apples-to- apples" comparison 
between AC and DC quantities that makes comparative circuit analysis much easier. 

The most popular standard of equivalence is based on work and power, and we call this the 
root-mean-square value of an AC waveform, or RMS for short. For example, an AC voltage of 120 
volts "RMS" means that this AC voltage is capable of producing the exact same amount of power 
(in Watts) at an electrical load as a 120 volt DC source powering the exact same load. 

The problem is exactly how to calculate this "RMS" value if all we know about the AC waveform 
is its peak value. If we compare a sine wave and a DC "wave" side by side, it is clear that the sine 
wave must peak at a greater value than the constant DC level in order to be equivalent in terms of 
doing the same work in the same amount of time: 



AC circuit 

DC circuit 




At first, it might seem like the correct approach would be to use calculus to integrate the sine 
wave over one-half of a cycle (from to 7r radians) and figure out how much area is under the 
curve. This is close, but not fully correct. You see, the ability of an electrical voltage to produce 
a power dissipation at a resistor is not directly proportional to the magnitude of that voltage, but 
rather proportional to the square of the magnitude of that voltage! In mathematical terms, power 
is predicted by the following equation: 



If we double the amount of voltage applied to a resistor, the power increases four-fold. If we triple 
the voltage, the power goes up by a factor of nine! If we are to figure out the "RMS" equivalent 
value of a sine wave, we must take this nonlinearity into consideration. 

First let us begin with a mathematical equivalence between the DC and AC cases. On one 
hand, the amount of work done by the DC voltage source will be equal to the power of that circuit 
multiplied by time. The unit of measurement for power is the Watt, which is defined as 1 Joule 


of work per second. So, multiplying the steady power rate in a DC circuit by the time we keep it 
powered will result in an answer of joules (total energy dissipated by the resistor): 

(V 2 
Work = — 

On the other hand, the amount of work done by a sine-wave-shaped AC voltage is equal to the 
square of the sine function divided by resistance, integrated over a specified time period. In other 
words, we will use the calculus process of integration to calculate the area underneath the function 
sin t rather than under the function sint. Since the interval from to 7r will encompass the essence 
of the sine wave's shape, this will be our integration interval: 

Work = / ^— dt 
Jo R 

Setting these two equations equal to each other (since we want the amount of work in each case 
to be equal), and making sure the DC side of the equation has -k for the amount of time (being the 
same interval as the AC side), we get this: 

V 2 \ rsm 2 t 

7T = / dt 

R ) Jo R 

First, we know that R is a constant value, and so we may move it out of the integrand: 

— 7r = — / sin t dt 
R ) RJo 

Multiplying both sides of the equation by R eliminates it completely. This should make intuitive 
sense, as our RMS equivalent value for a voltage is defined strictly by the ability to produce the 
same amount of power as the same value of DC voltage for any resistance value. Therefore the 
actual value of resistance (i?) should not matter, and it should come as no surprise that it falls out: 

V 2 ir = / sin 2 t dt 

Now, we may simplify the integrand by substituting the half-angle equivalence for the sin t 

, n l-cos2t , 

V 2 n = / dt 

Factoring one-half out of the integrand and moving it outside (because it's a constant): 

1 f* 

V 2 n = - l-cos2tdt 

2 Jo 

We may write this as the difference between two integrals, treating each term in the integrand 
as its own integration problem: 

V 2 TT = ~ [ I Idt- I COS 2/ (11 


The second integral may be solved simply by using substitution, with u = 2t, du = 2 dt, and 
dt= 4^: 


i / r , r cosw 

V A ix = - / 1 dt - I —— du 
Moving the one-half outside the second integrand: 

V tt = — I / 1 dt / cos u du 

2 V./o 2 Jo 

Finally, now we can integrate the silly thing: 

V" = ^IK-~\^K 


V 2 n = - ([tt - 0] - - [sin 2tt - sin 0] 

V 2 tt = - ( [tt-0] - -[0-0] 
V 2 ir= -(tt-0) 
V 2 tt = -tt 


We can see that tt cancels out of both sides: 

V 2 

Taking the square root of both sides, we arrive at our final answer: 

So, for a sinusoidal voltage with a peak value of 1 volt, the DC equivalent or "RMS" voltage 
value would be —?= volts, or approximately 0.707 volts. In other words, a sinusoidal voltage of 1 volt 
peak will produce just as much power dissipation at a resistor as a steady DC battery voltage of 
0.7071 volts applied to that same resistor. Therefore, this 1 volt peak sine wave may be properly 
called a 0.7071 volt RMS sine wave, or a 0.7071 volt "DC equivalent" sine wave. 

This factor for sinusoidal voltages is quite useful in electrical power system calculations, where 
the wave-shape of the voltage is nearly always sinusoidal (or very close). In your home, for example, 
the voltage available at any wall receptacle is 120 volts RMS, which translates to 169.7 volts peak. 

Electricians and electronics technicians often memorize the -4= conversion factor without realizing 
it only applies to sinusoidal voltage and current waveforms. If we are dealing with a non-sinusoidal 
wave-shape, the conversion factor between peak and RMS will be different! The mathematical 
procedure for obtaining the conversion factor will be identical, though: integrate the wave-shape's 
function (squared) over an interval sufficiently long to capture the essence of the shape, and set that 
equal to V 2 times that same interval span. 



4.2 Resistance, Reactance, and Impedance 

Resistance (R) is the dissipative opposition to an electric current, analogous to friction encountered 
by a moving object. Reactance (X) is the opposition to an electric current resulting from energy 
storage within circuit components, analogous to inertia of a moving object. Impedance (Z) is the 
combined total opposition to an electric current. 

Reactance comes in two opposing types: capacitive (Xq) and inductive (Xl). Each one is a 
function of frequency (/) in an AC circuit: 

X c 



X L = 2irfL 

4.3 Series and parallel circuits 

Impedance in a series circuit is the orthogonal sum of resistance and reactance: 


R 2 + (xl 


Equivalent series and parallel circuits are circuits that have the exact same total impedance 
as one another, one with series-connected resistance and reactance, and the other with parallel- 
connected resistance and reactance. The resistance and reactance values of equivalent series and 
parallel circuits may be expressed in terms of those circuits' total impedance: 









If the total impedance of one circuit (either series or parallel) is known, the component values of 
the equivalent circuit may be found by algebraically manipulating these equations and solving for 
the desired R and X values: 


series ^parallel 

x K 

series -^ parallel 



4.4 Phasor mathematics 

Something every beginning trigonometry student learns (or should learn) is how a sine wave is 
derived from the polar plot of a circle: 



-C — ' 

' ~~5~ 

%^ *\ o 


~^~—^ 1 

' ^^" 


C/2 n 






This translation from circular motion to a lengthwise plot has special significance to electrical 
circuits, because the circular diagram represents how alternating current (AC) is generated by a 
rotating machines, while the lengthwise plot shows how AC is generally displayed on a measuring 
instrument. The principle of an AC generator is that a magnet is rotated on a shaft past stationary 
coils of wire. When these wire coils experience the changing magnetic field produced by the rotating 
magnet, a sinusoidal voltage will be induced in the coils. 

- V coil 

Vcoil = V COS 6 

While sine and cosine wave graphs are quite descriptive, there is another type of graph that is 
even more descriptive for AC circuits: the so-called crank diagram. A "crank diagram" represents 



the sinusoidal wave not as a plot of instantaneous amplitude versus time, but rather as a plot of 
peak amplitude versus generator shaft angle. This is basically the polar-circular plot seen earlier, 
which beginning trigonometry students often see near the beginning of their studies: 




"" >. 




















IT i i 

^\ '' 



Direction of » 


vector rotation I 








By representing a sinusoidal voltage as a rotating vector instead of a graph over time, it is easier 
to see how multiple waveforms will interact with each other. Quite often in alternating-current (AC) 
circuits, we must deal with voltage waveforms that add with one another by virtue of their sources 
being connected in series. This sinusoidal addition becomes confusing if the two waveforms are not 
perfectly in step, which is often the case. However, out-of-step sinusoids are easy to represent and 
easy to sum when drawn as vectors in a crank diagram. Consider the following example, showing 
two sinusoidal waveforms, 60 degrees (? radians) out of step with each other: 





Graphically computing the sum of these two waves would be quite difficult in the standard graph 
(right-hand side), but it is as easy as stacking vectors tip-to-tail in the crank diagram: 


The length of the dashed-line vector A + B (radius of the dashed-line circle) represents the 
amplitude of the resultant sum waveform, while the phase shift is represented by the angles between 



this new vector and the original vectors A and B. 

This is all well and good, but we need to have a symbolic means of representing this same 
information if we are to do any real math with AC voltages and currents. There is one way to do 
this, if we take the leap of labeling the axes of the "crank diagram" as the axes of a complex plane 
(real and imaginary numbers): 





If we do this, we may symbolically represent each vector as a complex number. For example, 
vector B in the above diagram could be represented as the complex number x + jy (using j as the 
symbol for an imaginary quantity instead of i so as to not confuse it with current): 





Alternatively, we could express this complex quantity in polar form as an amplitude (B) and an 
angle (G), using the cosine and sine functions to translate this amplitude and angle into rectangular 



B(cos9+ jsinO) 

This is where things get really elegant. As you may recall, Euler's Relation relates complex 
exponential functions to trigonometric functions as such: 

e J = cos O + j sin 

With this critical piece of information, we have a truly elegant way to express all the information 
contained in the crank-diagram vector, in the form of an exponential term: 


In other words, this AC voltage, which is really a sinusoidal function over time, may be symbolized 
as a constant amplitude B (representing the peak voltage of the waveform) multiplied by a complex 
exponential (e-? ). What makes this representation really nice is that the complex exponential 
obeys all the mathematical laws we associate with real exponentials, including the differentiation 
and integration rules of calculus. This makes math operations much easier to deal with than if we 
had to represent AC voltages as trigonometric functions. 

Credit for this mathematical application goes to Charles Proteus Steinmetz, the brilliant electrical 
engineer (1865-1923). At the time, Steinmetz simply referred to this representation of AC waveforms 
as vectors. Now, we assign them the unique name of phasors so as to not confuse them with other 
types of vectors. The term "phasor" is quite appropriate, because the angle of a phasor (G) represents 
the phase shift between that waveform and a reference waveform. 

The notation has become so popular in electrical theory that even students who have never been 
introduced to Euler's Relation use them. In this case the notation is altered to make it easier to 
understand. Instead of writing Be^®, the mathematically innocent electronics student would write 


However, the real purpose of phasors is to make difficult math easier, so this is what we will 
explore now. Consider the problem of defining electrical opposition to current in an AC circuit. 
In DC (direct-current) circuits, resistance (R) is defined by Ohm's Law as being the ratio between 
voltage (V) and current (/): 

There are some electrical components, though, which do not obey Ohm's Law. Capacitors and 
inductors are two outstanding examples. The fundamental reason why these two components do not 
follow Ohm's Law is because they do not dissipate energy like resistances do. Rather than dissipate 
energy (in the form of heat and/or light), capacitors and inductors store and release energy from and 
to the circuit in which they are connected. The contrast between resistors and these components 
is remarkably similar to the contrast between friction and inertia in mechanical systems. Whether 
pushing a flat-bottom box across a floor or pushing a heavy wheeled cart across a floor, work is 
required to get the object moving. However, the flat-bottom box will immediately stop when you 
stop pushing it, while the wheeled cart will continue to coast because it has kinetic energy stored in 



The relationships between voltage and current for capacitors (C) and inductors (L) are as follows: 

I = C 


V = L- 



Expressed verbally, capacitors pass electric current proportional to how quickly the voltage across 
them changes over time. Conversely, inductors produce a voltage drop proportional to how quickly 
current through them changes over time. The symmetry here is beautiful: capacitors, which store 
energy in an electric field, oppose changes in voltage. Inductors, which store energy in a magnetic 
field, oppose changes in current. 

When either type of component is placed in an AC circuit, and subjected to sinusoidal voltages 
and currents, it will appear to have a "resistance." Given the amplitude of the circuit voltage and 
the frequency of oscillation (how rapidly the waveforms alternate over time), each type of component 
will only pass so much current. It would be nice, then, to be able to express the opposition each 
of these components offers to alternating current in the same way we express the resistance of a 
resistor in ohms (i7). To do this, we will have to figure out a way to take the above equations and 
manipulate them to express each component's behavior as a ratio of y. I will begin this process by 
using regular trigonometric functions to represent AC waveforms, then after seeing how ugly this 
gets I will switch to using phasors and you will see how much easier the math becomes. 

Let's start with the capacitor. Suppose we impress an AC voltage across a capacitor as such: 

AC voltage 
source {% 

V = sin cot 


It is common practice to represent the angle of an AC signal as the product u>t rather than as 
a static angle 0, with u> representing the angular velocity of the circuit in radians per second. If a 
circuit has a u) equal to 2ir, it means the generator shaft is making one full rotation every second. 
Multiplying oj by time t will give the generator's shaft position at that point in time. For example, 
in the United States our AC power grid operates at 60 cycles per second, or 60 revolutions of our 
ideal generator every second. This translates into an angular velocity u> of 120-7T radians per second, 
or approximately 377 radians per second. 

We know that the capacitor's relationship between voltage and current is as follows: 



Therefore, we may substitute the expression for voltage in this circuit into the equation and use 
calculus to differentiate it with respect to time: 

/ = C— (sinatf) 




The ratio of X- (the opposition to electric current, analogous to resistance R) will then be: 

V sin Lot 

I loC cos lot 





: tan lot 

This might look simple enough, until you realize that the ratio j 

' will become undefined for 
time-domain plot of voltage and current for 

certain values of t, notably | and 4^. If we look at 

a capacitor, it becomes clear why this is. There are points in time where voltage is maximum and 

current is zero: 

Max. voltage 
zero current 

Max. voltage, 
zero current 

Max. voltage, 
zero current 

At these instantaneous points in time, it truly does appear as if the "resistance" of the capacitor 
is undefined (infinite), with multiple incidents of maximum voltage and zero current. However, this 
does not capture the essence of what we are trying to do: relate the peak amplitude of the voltage 
with the peak amplitude of the current, to see what the ratio of these two peaks are. The ratio 
calculated here utterly fails because those peaks never happen at the same time. 

One way around this problem is to express the voltage as a complex quantity rather than as a 
scalar quantity. In other words, we use the sine and cosine functions to represent what this wave 
is doing, just like we used the "crank diagram" to represent the voltage as a rotating vector. By 
doing this, we can represent the waveforms as static amplitudes (vector lengths) rather than as 
instantaneous quantities that alternately peak and dip over time. The problem with this approach 
is that the math gets a lot tougher: 

I = C 


V = cos tot + j sin tot 

I = C — (cos cot + j sin Lot) 

C(—lo sin Lot + jlo cos ut) 


/ = ujC{— sin u)t + j cos u)t) 
V cos u>t + j sin u>t 

I uiC(— smut + j cosut) 

The final result is so ugly no one would want to use it. We may have succeeded in obtaining a 
ratio of V to I that doesn't blow up at certain values of t, but it provides no practical insight into 
what the capacitor will really do when placed in the circuit. 

Phasors to the rescue! Instead of representing the source voltage as a sum of trig functions 
(V = cosutt + j shoot), we will use Euler's Relation to represent it as a complex exponential and 
differentiate from there: 

I = C^- V = e^< 


I = c4- (e jut ) 

dt v ' 


Ce jwt 


e jut 








7 ' 



Note how the exponential term completely drops out of the equation, leaving us with a clean 
ratio strictly in terms of capacitance (C), angular velocity (u>), and of course j. This is the power 
of phasors: it transforms an ugly math problem into something trivial by comparison. 

Another detail of phasor math that is both beautiful and practical is the famous expression 
of Euler's Relation, the one all math teachers love because it directly relates several fundamental 
constants in one elegant equation: 

e™ = -1 

If you understand that this equation is nothing more than the fuller version of Euler's Relation 
with © set to the value of n, you may draw a few more practical insights from it: 

e* e = cos 6 + i sin 6 

COS 7T + t sin IT 

-1 + iO 



After seeing this, the natural question to ask is what happens when we set equal to other, 
common angles such as 0, ^, or ^? 

e i0 = cos + i sin 

e l ° = 1 + iO 

e l0 = l 

e ^ =cos (|) + , sin (|) 

.2) \2, 

e 2 = i 

3tt\ . /3tt 

e" 2 = cos I — + i sm — 
2 2 

e J ^ = - il 

We may show all the equivalencies on the complex plane, as unit vectors: 


e iJt/2 = i 

Q m = -1 


e l0 =l 


e i37t/2 = -i 



Going back to the result we got for the capacitor's opposition to current (y), we see that we can 
express the —i term (or —j term, as it is more commonly written in electronics work) as a complex 
exponential and gain a little more insight: 

V _ .1 

7 ~~ J ZJC 

V / 13*\ 1 

I V e : J LUC 

What this means is that the capacitor's opposition to current takes the form of a phasor pointing 
down on the complex plane. In other words, it is a phasor with a fixed angle (4^, or — ^ radians) 
rather than rotating around the origin like all the voltage and current phasors do. In electric circuit 
theory, there is a special name we give to such a quantity, being a ratio of voltage to current, 
but possessing a complex value. We call this quantity impedance rather than resistance, and we 
symbolize it using the letter Z. 

When we do this, we arrive at a new form of Ohm's Law for AC circuits: 

V V 

Z = — V = IZ I = — 

I Z 

With all quantities expressed in the form of phasors, we may apply nearly all the rules of DC 
circuits (Ohm's Law, Kirchhoff's Laws, etc.) to AC circuits. What was old is new again! 


Boylestad, Robert L., Introductory Circuit Analysis, 9th Edition, Prentice Hall, Upper Saddle River, 
New Jersey, 2000. 

Steinmetz, Charles P., Theory and Calculation of Alternating Current Phenomena, Third Edition, 
McGraw Publishing Company, New York, NY, 1900. 


Chapter 5 

Introduction to Industrial 

Instrumentation is the science of automated measurement and control. Applications of this science 
abound in modern research, industry, and everyday living. From automobile engine control systems 
to home thermostats to aircraft autopilots to the manufacture of pharmaceutical drugs, automation 
surrounds us. This chapter explains some of the fundamental principles of industrial instrumentation. 

The first step, naturally, is measurement. If we can't measure something, it is really pointless to 
try to control it. This "something" usually takes one of the following forms in industry: 

• Fluid pressure 

• Fluid flow rate 

• The temperature of an object 

• Fluid volume stored in a vessel 

• Chemical concentration 

• Machine position, motion, or acceleration 

• Physical dimension(s) of an object 

• Count (inventory) of objects 

• Electrical voltage, current, or resistance 

Once we measure the quantity we are interested in, we usually transmit a signal representing 
this quantity to an indicating or computing device where either human or automated action then 
takes place. If the controlling action is automated, the computer sends a signal to a final controlling 
device which then influences the quantity being measured. This final control device usually takes 
one of the following forms: 




• Control valve (for throttling the flow rate of a fluid) 

• Electric motor 

• Electric heater 

Both the measurement device and the final control device connect to some physical system which 
we call the process. To show this as a general block diagram: 






Final control 


The Process 

The common home thermostat is an example of a measurement and control system, with the 
home's internal air temperature being the "process" under control. In this example, the thermostat 
usually serves two functions: sensing and control, while the home's heater adds heat to the home 
to increase temperature, and/or the home's air conditioner extracts heat from the home to decrease 
temperature. The job of this control system is to maintain air temperature at some comfortable 
level, with the heater or air conditioner taking action to correct temperature if it strays too far from 
the desired value (called the setpoint). 

Industrial measurement and control systems have their own unique terms and standards, which is 
the primary focus of this lesson. Here are some common instrumentation terms and their definitions: 

Process: The physical system we are attempting to control or measure. Examples: water filtration 
system, molten metal casting system, steam boiler, oil refinery unit, power generation unit. 

Process Variable, or PV: The specific quantity we are measuring in a process. Examples: pressure, 
level, temperature, flow, electrical conductivity, pH, position, speed, vibration. 

Setpoint, or SP: The value at which we desire the process variable to be maintained at. In other 
words, the "target" value of the process variable. 


Primary Sensing Element, or PSE: A device that directly senses the process variable and 
translates that sensed quantity into an analog representation (electrical voltage, current, resistance; 
mechanical force, motion, etc.). Examples: thermocouple, thermistor, bourdon tube, microphone, 
potentiometer, electrochemical cell, accelerometer. 

Transducer: A device that converts one standardized instrumentation signal into another 
standardized instrumentation signal, and/or performs some sort of processing on that signal. 
Examples: I/P converter (converts 4-20 mA electric signal into 3-15 PSI pneumatic signal), P/I 
converter (converts 3-15 PSI pneumatic signal into 1^-20 mA electric signal), square-root extractor 
(calculates the square root of the input signal). 

Note: in general science parlance, a "transducer" is any device that converts one form of energy 
into another, such as a microphone or a thermocouple. In industrial instrumentation, however, we 
generally use "primary sensing element" to describe this concept and reserve the word "transducer" 
to specifically refer to a conversion device for standardized instrumentation signals. 

Transmitter: A device that translates the signal produced by a primary sensing element (PSE) into 
a standardized instrumentation signal such as 3-15 PSI air pressure, 4-20 mA DC electric current, 
Fieldbus digital signal packet, etc., which may then be conveyed to an indicating device, a controlling 
device, or both. 

Lower- and Upper-range values, abbreviated LRV and URV, respectively: the values of process 
measurement deemed to be 0% and 100% of a transmitter's calibrated range. For example, if a 
temperature transmitter is calibrated to measured a range of temperature starting at 300 degrees 
Celsius and ending at 500 degrees Celsius, 300 degrees would be the LRV and 500 degrees the URV. 

Controller: A device that receives a process variable (PV) signal from a primary sensing element 
(PSE) or transmitter, compares that signal to the desired value for that process variable (called the 
setpoint), and calculates an appropriate output signal value to be sent to a final control element 
(FCE) such as an electric motor or control valve. 

Final Control Element, or FCE: A device that receives the signal from a controller to directly 
influence the process. Examples: variable-speed electric motor, control valve, electric heater. 

Automatic mode: When the controller generates an output signal based on the relationship of 
process variable (PV) to the setpoint (SP). 

Manual mode: When the controller's decision-making ability is bypassed to let a human operator 
directly determine the output signal sent to the final control element. 

Now I will show you some practical examples of measurement and control systems so you can 
get a better idea of these fundamental concepts. 


5.1 Example: boiler water level control system 

Steam boilers are very common in industry, principally because steam power is so useful. Common 
uses for steam in industry include doing mechanical work (e.g. a steam engine moving some sort 
of machine), heating, producing vacuums (through the use of "steam eductors"), and augmenting 
chemical processes (e.g. reforming of natural gas into hydrogen and carbon dioxide). 

The process of converting water into steam is quite simple: heat up the water until it boils. 
Anyone who has ever boiled a pot of water for cooking knows how this process works. Making 
steam continuously, however, is a little more complicated. The fundamental variable to measure 
and control in a continuous boiler is the level of water in the "steam drum" (the upper vessel in a 
water-tube boiler). In order to safely and efficiently produce a continuous flow of steam, we must 
ensure the steam drum never runs too low on water, or too high. If there is not enough water in 
the drum, the water tubes may run dry and burn through from the heat of the fire. If there is too 
much water in the drum, liquid water may be carried along with the flow of steam, causing problems 

In this next illustration, you can see the essential elements of a water level control system, 
showing transmitter, controller, and control valve: 



Steam drum water level control 
system for an industrial boiler A s - 

Exhaust stack 






3-15 PSI 

The first instrument in this control system is the level transmitter, or "LT" . The purpose of this 
device is to sense the water level in the steam drum and report that measurement to the controller 
in the form of an instrument signal. In this case, the type of signal is pneumatic: a variable air 
pressure sent through metal or plastic tubes. The greater the water level in the drum, the more air 
pressure output by the level transmitter. Since the transmitter is pneumatic, it must be supplied 
with a source of clean, compressed air on which to run. This is the meaning of the "A.S." tube (Air 
Supply) entering the top of the transmitter. 

This pneumatic signal is sent to the next instrument in the control system, the level indicating 
controller, or "LIC". The purpose of this instrument is to compare the level transmitter's signal 
with a setpoint value entered by a human operator (the desired water level in the steam drum). The 
controller then generates an output signal telling the control valve to either introduce more or less 
water into the boiler to maintain the steam drum water level at setpoint. As with the transmitter, 
the controller in this system is pneumatic, operating entirely on compressed air. This means the 
output of the controller is also a variable air pressure signal, just like the signal output by the level 
transmitter. Naturally, the controller requires a constant supply of clean, compressed air on which 


to run, which explains the "A.S." (Air Supply) tube connecting to it. 

The last instrument in this control system is the control valve, being operated directly by the air 
pressure signal generated by the controller. This particular control valve uses a large diaphragm to 
convert the air pressure signal into a mechanical force to move the valve open and closed. A large 
spring inside the valve mechanism provides the force necessary to return the valve to its normal 
position, while the force generated by the air pressure on the diaphragm works against the spring 
to move the valve the other direction. 

When the controller is placed in the "automatic" mode, it will move the control valve to whatever 
position it needs to be in order to maintain a constant steam drum water level. The phrase "whatever 
position it needs to be" suggests that the relationship between the controller output signal, the 
process variable signal (PV), and the setpoint (SP) can be quite complex. If the controller senses a 
water level above setpoint, it will take whatever action is necessary to bring that level back down 
to setpoint. Conversely, if the controller senses a water level below setpoint, it will take whatever 
action is necessary to bring that level up to setpoint. What this means in a practical sense is that 
the controller's output signal (equating to valve position) is just as much a function of process load 
(i.e. how much steam is being used from the boiler) as it is a function of setpoint. 

Consider a situation where the steam demand from the boiler is very low. If there isn't much 
steam being drawn off the boiler, this means there will be little water boiled into steam and therefore 
little need for additional feedwater to be pumped into the boiler. Therefore, in this situation, one 
would expect the control valve to hover near the fully-closed position, allowing just enough water 
into the boiler to keep the steam drum water level at setpoint. 

If, however, there is great demand for steam from this boiler, the rate of evaporation will be 
much higher. This means the control system will have to add feedwater to the boiler at a much 
greater flow rate in order to maintain the steam drum water level at setpoint. In this situation we 
would expect to see the control valve much closer to being fully-open as the control system "works 
harder" to maintain a constant water level in the steam drum. 

A human operator running this boiler has the option of placing the controller into "manual" 
mode. In this mode, the control valve position is under direct control of the human operator, 
with the controller essentially ignoring the signal sent from the water level transmitter. Being an 
indicating controller, the controller faceplate will still show how much water is in the steam drum, 
but it is now the human operator's sole responsibility to move the control valve to the appropriate 
position to hold water level at setpoint. 

Manual mode is useful to the human operator(s) during start-up and shut-down conditions. It is 
also useful to the instrument technician for troubleshooting a mis-behaving control system. When a 
controller is in automatic mode, the output signal (sent to the control valve) changes in response to 
the process variable (PV) and setpoint (SP) values. Changes in the control valve position, in turn, 
naturally affect the process variable signal through the physical relationships of the process. What 
we have here is a situation where causality is uncertain. If we see the process variable changing 
erratically over time, does this mean we have a faulty transmitter (outputting an erratic signal), or 
does it mean the controller output is erratic (causing the control valve to shift position unnecessarily) , 
or does it mean the steam demand is fluctuating and causing the water level to vary as a result? 
So long as the controller remains in automatic mode, we can never be completely sure what is 
causing what to happen, because the chain of causality is actually a loop, with everything affecting 
everything else in the system. 



A simple way to diagnose such a problem is to place the controller in manual mode. Now the 
output signal to the control valve will be fixed at whatever level the human operator or technician 
sets it to. If we see the process variable signal suddenly stabilize, we know the problem has something 
to do with the controller output. If we see the process variable signal suddenly become even more 
erratic once we place the controller in manual mode, we know the controller was actually trying to 
do its job properly in automatic mode and the cause of the problem lies within the process itself. 

As was mentioned before, this is an example of a pneumatic (compressed air) control system, 
where all the instruments operate on compressed air, and use compressed air as the signaling medium. 
Pneumatic instrumentation is an old technology, dating back many decades. While most modern 
instruments are electronic in nature, pneumatic instruments still find application within industry. 
The most common industry standard for pneumatic pressure signals is 3 to 15 PSI, with 3 PSI 
representing low end-of-scale and 15 PSI representing high end-of-scale. The following table shows 
the meaning of different signal pressures are they relate to the level transmitter's output: 

Transmitter air signal pressure 

Steam drum water level 

3 PSI 

0% (Empty) 

6 PSI 


9 PSI 


12 PSI 


15 PSI 

100% (Full) 

Likewise, the controller's pneumatic output signal to the control valve uses the same 3 to 15 PSI 
standard to command different valve positions: 

Controller output signal pressure 

Control valve position 

3 PSI 

0% open (Fully shut) 

6 PSI 

25% open 

9 PSI 

50% open 

12 PSI 

75% open 

15 PSI 

100% (Fully open) 

It should be noted the previously shown transmitter calibration table assumes the transmitter 
measures the full range of water level possible in the drum. Usually, this is not the case. Instead, 
the transmitter will be calibrated so that it only senses a narrow range of water level near the middle 
of the drum. Thus, 3 PSI (0%) will not represent an empty drum, and neither will 15 PSI (100%) 
represent a completely full drum. Calibrating the transmitter like this helps avoid the possibility of 
actually running the drum completely empty or completely full in the case of an operator incorrectly 
setting the setpoint value near either extreme end of the measurement scale. 

An example table showing this kind of realistic transmitter calibration is shown here: 



Transmitter air signal pressure 

Actual steam drum water level 

3 PSI 


6 PSI 


9 PSI 


12 PSI 


15 PSI 




5.2 Example: wastewater disinfection 

The final step in treating wastewater before releasing it into the natural environment is to kill any 
harmful bacteria in it. This is called disinfection, and chlorine gas is a very effective disinfecting 
agent. However, just as it is not good to mix too little chlorine in the outgoing water (effluent) 
because we might not disinfect the water thoroughly enough, there is also danger of injecting too 
much chlorine in the effluent because then we might begin poisoning animals and beneficial micro- 
organisms in the natural environment. 

To ensure the right amount of chlorine injection, we must use a dissolved chlorine analyzer to 
measure the chlorine concentration in the effluent, and use a controller to automatically adjust 
the chlorine control valve to inject the right amount of chlorine at all times. The following P&ID 
(Process and Instrument Diagram) shows how such a control system might look: 

Chlorine supply 


control valve 

4-20 mA 


Influent h 


4-20 mA 
measurement i 





-> Effluent 

Chlorine gas coming through the control valve mixes with the incoming water (influent), then 
has time to disinfect in the contact chamber before exiting out to the environment. 

The transmitter is labeled "AT" (Analytical Transmitter) because its function is to analyze the 
concentration of chlorine dissolved in the water and transmit this information to the control system. 
The "CI2" written near the transmitter bubble declares this to be a chlorine analyzer. The dashed 
line coming out of the transmitter tells us the signal is electronic in nature, not pneumatic as was 
the case in the previous (boiler control system) example. The most common and likely standard for 
electronic signaling in industry is 4 to 20 milliamps DC, which represents chlorine concentration in 
much the same way as the 3 to 15 PSI pneumatic signal standard represented steam drum water 
level in the previous system: 

Transmitter signal current 

Chlorine concentration 

4 mA 

0% (no chlorine) 

8 mA 


12 mA 


16 mA 


20 mA 

100% (Full concentration) 



The controller is labeled "AIC" because it is an Analytical Indicating Controller. Controllers are 
always designated by the process variable they are charged with controlling, in this case the chlorine 
analysis of the effluent. "Indicating" means there is some form of display that a human operator or 
technician can read showing the chlorine concentration. "SP" refers to the setpoint value entered by 
the operator, which the controller tries to maintain by adjusting the position of the chlorine injection 

A dashed line going from the controller to the valve indicates another electronic signal, most likely 
4 to 20 mA DC again. Just as with the 3 to 15 PSI pneumatic signal standard in the pneumatic 
boiler control system, the amount of electric current in this signal path directly relates to a certain 
valve position: 

Controller output signal current 

Control valve position 

4 mA 

0% open (Fully shut) 

8 mA 

25% open 

12 mA 

50% open 

16 mA 

75% open 

20 mA 

100% (Fully open) 

Note: it is possible, and in some cases even preferable, to have either a transmitter or a control 
valve that responds in reverse fashion to an instrument signal such as 3 to 15 PSI or 4 to 20 milliamps. 
For example, this valve could have been set up to be wide open at 4 mA and fully shut at 20 mA. 
The main point to recognize here is that both the process variable sensed by the transmitter and 
the position of the control valve are proportionately represented by an analog signal. 

The letter "M" inside the control valve bubble tells us this is a motor-actuated valve. Instead 
of using compressed air pushing against a spring-loaded diaphragm as was the case in the boiler 
control system, this valve is actuated by an electric motor turning a gear-reduction mechanism. The 
gear reduction mechanism allows slow motion of the control valve stem even though the motor spins 
at a fast rate. A special electronic control circuit inside the valve actuator modulates electric power 
to the electric motor in order to ensure the valve position accurately matches the signal sent by the 
controller. In effect, this is another control system in itself, controlling valve position according to a 
"setpoint" signal sent by another device (in this case, the AIT controller which is telling the valve 
what position to go to). 


5.3 Example: chemical reactor temperature control 

Sometimes we see a mix of instrument signal standards in one control system. Such is the case for this 
particular chemical reactor temperature control system, where three different signal standards are 
used to convey information between the instruments. A P&ID (Process and Instrument Diagram) 
shows the inter-relationships of the process piping, vessels, and instruments: 

3-15 PSI 

-H- H- 


1 Feed in 




Fieldbus (digital) 



4-20 mA 
control /- r , r ~ 
signal ( I IU 


$— i Product out 

The purpose of this control system is to ensure the chemical solution inside the reactor vessel 
is maintained at a constant temperature. A steam-heated "jacket" envelops the reactor vessel, 
transferring heat from the steam into the chemical solution inside. The control system maintains 
a constant temperature by measuring the temperature of the reactor vessel, and throttling steam 
from a boiler to the steam jacket to add more or less heat as needed. 

We begin as usual with the temperature transmitter, located near the bottom of the vessel. Note 
the different line type used to connect the temperature transmitter (TT) with the temperature- 
indicating controller (TIC): solid dots with lines in between. This signifies a digital electronic 
instrument signal - sometimes referred to as a fieldbus - rather than an analog type (such as 4 
to 20 mA or 3 to 15 PSI). The transmitter in this system is actually a computer, and so is the 
controller. The transmitter reports the process variable (reactor temperature) to the controller 
using digital bits of information. Here there is no analog scale of 4 to 20 milliamps, but rather 
electric voltage/current pulses representing the and 1 states of binary data. 

Digital instrument signals are not only capable of transferring simple process data, but they 
can also convey device status information (such as self-diagnostic test results). In other words, the 


digital signal coming from this transmitter not only tells the controller how hot the reactor is, but 
it can also tell the controller how well the transmitter is functioning! 

The dashed line exiting the controller shows it to be analog electronic: most likely 4 to 20 
milliamps DC. This electronic signal does not go directly to the control valve, however. It passes 
through a device labeled "TY" , which is a transducer to convert the 4 to 20 niA electronic signal 
into a 3 to 15 PSI pneumatic signal which then actuates the valve. In essence, this signal transducer 
acts as an electrically-controlled air pressure regulator, taking the supply air pressure (usually 20 to 
25 PSI) and regulating it down to a level commanded by the controller's electronic output signal. 

At the temperature control valve (TV) the 3 to 15 PSI pneumatic pressure signal applies a force 
on a diaphragm to move the valve mechanism against the restraining force of a large spring. The 
construction and operation of this valve is the same as for the feedwater valve in the pneumatic 
boiler water control system. 



5.4 Other types of instruments 

So far we have just looked at instruments that sense, control, and influence process variables. 
Transmitters, controllers, and control valves are respective examples of each instrument type. 
However, other instruments exist to perform useful functions for us. 

One common "auxiliary" instrument is the indicator, the purpose of which is to provide a human- 
readable indication of an instrument signal. Quite often process transmitters are not equipped with 
readouts for whatever variable they measure: they just transmit a standard instrument signal (3 to 
15 PSI, 4 to 20 mA, etc.) to another device. An indicator gives a human operator a convenient 
way of seeing what the output of the transmitter is without having to connect test equipment 
(pressure gauge for 3-15 PSI, ammeter for 4-20 mA) and perform conversion calculations. Moreover, 
indicators may be located far from their respective transmitters, providing readouts in locations 
more convenient than the location of the transmitter itself. An example where remote indication 
would be practical is shown here, in a nuclear reactor temperature measurement system: 



4-20 mA signal 




No human can survive inside the concrete-walled containment vessel when the nuclear reactor 
is operating, due to the strong radiation flux around the reactor. The temperature transmitter is 
built to withstand the radiation, though, and it transmits a 4 to 20 milliamp electronic signal to an 



indicating recorder located outside of the containment building where it is safe for a human operator 
to be. There is nothing preventing us from connecting multiple indicators, at multiple locations, 
to the same 4 to 20 milliamp signal wires coming from the temperature transmitter. This allows 
us to display the reactor temperature in as many locations as we desire, since there is no absolute 
limitation on how far we may conduct a DC milliamp signal along copper wires. 

Another common "auxiliary" instrument is the recorder (sometimes specifically referred to as a 
chart recorder or a trend recorder), the purpose of which is to draw a graph of process variable(s) 
over time. Recorders usually have indications built into them for showing the instantaneous value 
of the instrument signal(s) simultaneously with the historical values, and for this reason are usually 
designated as indicating recorders. A temperature indicating recorder for the nuclear reactor system 
shown previously would be designated as a "TIR" accordingly. 

Recorders are extremely helpful for troubleshooting process control problems. This is especially 
true when the recorder is configured to record not just the process variable, but also the controller's 
setpoint and output variables as well. Here is an example of a typical "trend" showing the 
relationship between process variable, setpoint, and controller output in automatic mode, as graphed 
by a recorder: 








- r- 









Time — *- 

Here, the setpoint (SP) appears as a perfectly straight (red) line, the process variable as a slightly 
bumpy (blue) line, and the controller output as a very bumpy (purple) line. We can see from this 
trend that the controller is doing exactly what it should: holding the process variable value close to 
setpoint, manipulating the final control element as far as necessary to do so. The erratic appearance 
of the output signal is not really a problem, contrary to most peoples' first impression. The fact 
that the process variable never deviates significantly from the setpoint tells us the control system is 
operating quite well. What accounts for the erratic controller output, then? Variations in process 
load. As other variables in the process vary, the controller is forced to compensate for these variations 
in order that the process variable does not drift from setpoint. Now, maybe this does indicate a 
problem somewhere else in the process, but there is certainly no problem in this control system. 



Recorders become powerful diagnostic tools when coupled with the controller's manual control 
mode. By placing a controller in "manual" mode and allowing direct human control over the final 
control element (valve, motor, heater), we can tell a lot about a process. Here is an example of a 
trend recording for a process in manual mode, where the process variable response is seen graphed 
in relation to the controller output as that output is increased and decreased in steps: 



^_,^^^^,^^« .— ^ 

D\/ S \ 

/ ^-^ 

Oi itn it 


Notice the time delay between when the output signal is "stepped" to a new value and when 
the process variable responds to the change. This sort of delay is generally not good for a control 
system. Imagine trying to steer an automobile whose front wheels respond to your input at the 
steering wheel only after a 5-second delay! This would be a very challenging car to drive, because 
the steering is grossly delayed. The same problem plagues any industrial control system with a time 
lag between the final control element and the transmitter. Typical causes of this problem include 
transport delay (where there is a physical delay resulting from transit time of a process medium 
from the point of control to the point of measurement) and mechanical problems in the final control 

This next example shows another type of problem revealed by a trend recording during manual- 
mode testing: 







I. ... 










Time — *- 

Here, we see the process quickly responding to all step-changes in controller output except for 
those involving a change in direction. This problem is usually caused by mechanical friction in the 
final control element (e.g. sticky valve stem packing in a pneumatically-actuated control valve), and 
is analogous to "loose" steering in an automobile, where the driver must turn the steering wheel a 
little bit extra after reversing steering direction. Anyone who has ever driven an old farm tractor 
knows what this phenomenon is like, and how it detrimentally affects one's ability to steer the tractor 
in a straight line. 

Another type of instrument commonly seen in measurement and control systems is the process 
switch. The purpose of a switch is to turn on and off with varying process conditions. Usually, 
switches are used to activate alarms to alert human operators to take special action. In other 
situations, switches are directly used as control devices. The following P&ID of a compressed air 
control system shows both uses of process switches: 








The "PSH" (pressure switch, high) activates when the air pressure inside the vessel reaches its 
high control point. The "PSL" (pressure switch, low) activates when the air pressure inside the 
vessel drops down to its low control point. Both switches feed discrete (on/off) electrical signals to 
a logic control device (signified by the diamond) which then controls the starting and stopping of 
the electric motor-driven air compressor. 

Another switch in this system labeled "PSHH" (pressure switch, high-high) activates only if 
the air pressure inside the vessel exceeds a level beyond the high shut-off point of the high pressure 
control switch (PSH). If this switch activates, something has gone wrong with the compressor control 
system, and the high pressure alarm (PAH, or pressure alarm, high) activates to notify a human 

All three switches in this air compressor control system are directly actuated by the air pressure 
in the vessel. In other words these are process-sensing switches. It is possible to build switch devices 
that interpret standardized instrumentation signals such as 3 to 15 PSI (pneumatic) or 4 to 20 
milliamps (analog electronic), which allows us to build on/off control systems and alarms for any 
type of process variable we can measure with a transmitter. For example, the chlorine wastewater 
disinfection system shown earlier may be equipped with a couple of alarm switches to alert an 
operator if the chlorine concentration ever exceeds pre-determined high or low limits: 



Chlorine supply 

Influent h 









-> Effluent 

The labels "AAL" and "AAH" refer to analytical alarm low and analytical alarm high, 
respectively. Since both alarms work off the 4 to 20 milliamp electronic signal output by the chlorine 
analytical transmitter (AT) rather than directly sensing the process, their construction is greatly 
simplified. If these were process-sensing switches, each one would have to be equipped with the 
capability of directly sensing chlorine concentration. In other words, each switch would have to be 
its own chlorine concentration analyzer, with all the inherent complexity of such a device! 

5.5 Summary 

Instrument technicians maintain the safe and efficient operation of industrial measurement and 
control systems. As this chapter shows, this requires a broad command of technical skill. 
Instrumentation is more than just physics or chemistry or mathematics or electronics or mechanics 
or control theory alone. An instrument technician must understand all these subject areas to some 
degree, and more importantly how these knowledge areas relate to each other. 

The all-inclusiveness of this profession makes it very challenging and interesting. Adding to the 
challenge is the continual introduction of new technologies. The advent of new technologies, however, 
does not necessarily relegate legacy technologies to the scrap heap. It is quite common to find state- 
of-the-art instruments in the very same facility as decades-old instruments; digital fieldbus networks 
running alongside 3 to 15 PSI pneumatic signal tubes; microprocessor-based sensors mounted right 
next to old mercury tilt-switches. Thus, the competent instrument technician must be comfortable 
working with both old and new technologies, understanding the relative merits and weaknesses of 

This is why the most important skill for an instrument technician is the ability to teach oneself. 
It is impossible to fully prepare for a career like this with any amount of preparatory schooling. The 
profession is so broad and the responsibility so great, and the landscape so continuously subject to 
change, that life-long learning for the technician is a matter of professional survival. 

Chapter 6 

Instrumentation documents 

Every technical discipline has its own standardized way(s) of making descriptive diagrams, and 
instrumentation is no exception. The scope of instrumentation is so broad, however, that no one 
form of diagram is sufficient to capture all we might need to represent. This chapter will discuss 
three different types of instrumentation diagrams: 



Process Flow Diagrams (PFDs) 
Process and Instrument diagrams (P&IDs) 
• Loop diagrams 
SAMA diagrams 


At the highest level, the instrument technician is interested in the interconnections of process 
vessels, pipes, and flow paths of process fluids. The proper form of diagram to represent the "big 
picture" of a process is called a process flow diagram. Individual instruments are sparsely represented 
in a PFD, because the focus of the diagram is the process itself. 

At the lowest level, the instrument technician is interested in the interconnections of individual 
instruments, including all the wire numbers, terminal numbers, cable types, instrument calibration 
ranges, etc. The proper form of diagram for this level of fine detail is called a loop diagram. Here, 
the process vessels and piping are sparsely represented, because the focus of the diagram is the 
instruments themselves. 

Process and instrument diagrams (P&IDs) lie somewhere in the middle between process flow 
diagrams and loop diagrams. A P&ID shows the layout of all relevant process vessels, pipes, and 
machinery, but with instruments superimposed on the diagram showing what gets measured and 
what gets controlled. Here, one can view the flow of the process as well as the "flow" of information 
between instruments measuring and controlling the process. 

SAMA diagrams are used for an entirely different purpose: to document the strategy of a control 
system. In a SAMA diagram, emphasis is placed on the algorithms used to control a process, as 
opposed to piping, wiring, or instrument connections. These diagrams are commonly found within 
the power generation industry, but are sometimes used in other industries as well. 



An instrument technician must often switch between different diagrams when troubleshooting a 
complex control system. There is simply too much detail for any one diagram to show everything. 
Even if the page were large enough, a "show everything" diagram would be so chock-full of details 
that it would be difficult to follow any one line of details you happened to be interested in at any 
particular time. The narrowing of scope with the progression from PFD to loop diagram may be 
visualized as a process of "zooming in," as though one were viewing a process through the lens of 
a microscope at different powers. First you begin with a PFD or P&ID to get an overview of the 
process, to see how the major components interact. Then, once you have identified which instrument 
"loop" you need to investigate, you go to the appropriate loop diagram to see the interconnection 
details of that instrument system so you know where to connect your test equipment and what 
signals you expect to find when you do. 

Another analogy for this progression of documents is a map, or more precisely, a globe, an 
atlas, and a city street map. The globe gives you the "big picture" of the Earth, countries, and 
major cities. An atlas allows you to "zoom in" to see details of particular provinces, states, and 
principalities, and the routes of travel connecting them all. A city map shows you major and minor 
roads, canals, alleyways, and perhaps even some addresses in order for you to find your way to a 
particular destination. It would be impractical to have a globe large enough to show you all the 
details of every city! Furthermore, a globe comprehensive enough to show you all these details 
would have to be updated very frequently to keep up with all cities' road changes. There is a certain 
economy inherent to the omission of fine details, both in ease of use and in ease of maintenance. 



6.1 Process Flow Diagrams 

To show a practical process example, let's examine three diagrams for a compressor control system. 
In this fictitious process, water is being evaporated from a process solution under partial vacuum 
(provided by the compressor). The compressor then transports the vapors to a "knockout drum" 
where some of them condense into liquid form. As a typical PFD, this diagram shows the major 
interconnections of process vessels and equipment, but omits details such as instrument signal lines 
and auxiliary instruments: 


*[><] — > Water 

> Condensate 
> Brine 

One might guess the instrument interconnections based on the instruments' labels. For instance, 
a good guess would be that the level transmitter (LT) on the bottom of the knockout drum might 
send the signal that eventually controls the level valve (LV) on the bottom of that same vessel. One 
might also guess that the temperature transmitter (TT) on the top of the evaporator might be part 
of the temperature control system that lets steam into the heating jacket of that vessel. 

Based on this diagram alone, one would be hard-pressed to determine what control system, if 


any, controls the compressor itself. All the PFD shows relating directly to the compressor is a flow 
transmitter (FT) on the suction line. This level of uncertainty is perfectly acceptable for a PFD, 
because its purpose is merely to show the general flow of the process itself, and only a bare minimum 
of control instrumentation. 


6.2 Process and Instrument Diagrams 

The next level of detail is the Process and Instrument Diagram, or P&ID. Here, we see a "zooming 
in" of scope from the whole evaporator process to the compressor as a unit. The evaporator and 
knockout vessels almost fade into the background, with their associated instruments absent from 

Now we see there is more instrumentation associated with the compressor than just a flow 
transmitter. There is also a differential pressure transmitter (PDT), a flow indicating controller 
(FIC), and a "recycle" control valve that allows some of the vapor coming out of the compressor's 
discharge line to go back around into the compressor's suction line. Additionally, we have a pair 
of temperature transmitters that report suction and discharge line temperatures to an indicating 

Some other noteworthy details emerge in the P&ID as well. We see that the flow transmitter, flow 
controller, pressure transmitter, and flow valve all bear a common number: 42. This common "loop 
number" indicates these four instruments are all part of the same control system. An instrument 


with any other loop number is part of a different control system, measuring and/or controlling some 
other function in the process. Examples of this include the two temperature transmitters and their 
respective recorders, bearing the loop numbers 41 and 43. 

Please note the differences in the instrument "bubbles" as shown on this P&ID. Some of the 
bubbles are just open circles, where others have lines going through the middle. Each of these 
symbols has meaning according to the ISA (Instrumentation, Systems, and Automation society) 

Panel-mounted Panel-mounted 

Field-mounted (main control room) (auxiliary location) 

Front of panel Front of panel 

Rear of panel Rear of panel 

The type of "bubble" used for each instrument tells us something about its location. This, 
obviously, is quite important when working in a facility with many thousands of instruments scattered 
over acres of facility area, structures, and buildings. 

The rectangular box enclosing both temperature recorders shows they are part of the same 
physical instrument. In other words, this indicates there is really only one temperature recorder 
instrument, and that it plots both suction and discharge temperatures (most likely on the same 
trend graph). This suggests that each bubble may not necessarily represent a discrete, physical 
instrument, but rather an instrument function that may reside in a multi-function device. 

Details we do not see on this P&ID include cable types, wire numbers, terminal blocks, junction 
boxes, instrument calibration ranges, failure modes, power sources, and the like. To examine this 
level of detail, we must go to the loop diagram we are interested in. 



6.3 Loop diagrams 

Finally, we arrive at the loop diagram (sometimes called a loop sheet) for the compressor surge 
control system (loop number 42): 

Loop Diagram: Compressor surge control 

Here we see that the P&ID didn't show us all the instruments in this control "loop." Not only do 
we have two transmitters, a controller, and a valve; we also have two signal transducers. Transducer 
42a modifies the flow transmitter's signal before it goes into the controller, and transducer 42b 
converts the electronic 4 to 20 mA signal into a pneumatic 3 to 15 PSI air pressure signal. Each 
instrument "bubble" in a loop diagram represents an individual device, with its own terminals for 
connecting wires. 

Note that dashed lines now represent individual copper wires instead of whole cables. Terminal 
blocks where these wires connect to are represented by squares with numbers in them. Cable 
numbers, wire colors, junction block numbers, panel identification, and even grounding points are 
all shown in loop diagrams. The only type of diagram at a lower level of abstraction than a loop 
diagram would be an electronic schematic diagram for an individual instrument, which of course 


would only show details pertaining to that one instrument. Thus, the loop diagram is the most 
detailed form of diagram for a control system as a whole, and thus it must contain all details 
omitted by PFDs and P&IDs alike. 

To the novice it may seem excessive to include such trivia as wire colors in a loop diagram. To 
the experienced instrument technician who has had to work on systems lacking such documented 
detail, this information is highly valued. The more detail you put into a loop diagram, the easier 
it makes the inevitable job of maintaining that system at some later date. When a loop diagram 
shows you exactly what wire color to expect at exactly what point in an instrumentation system, 
and exactly what terminal that wire should connect to, it becomes much easier to proceed with any 
troubleshooting, calibration, or upgrade task. 

An interesting detail seen on this loop diagram is an entry specifying "input calibration" and 
"output calibration" for each and every instrument in the system. This is actually a very important 
concept to keep in mind when troubleshooting a complex instrumentation system: every instrument 
has at least one input and at least one output, with some sort of mathematical relationship between 
the two. Diagnosing where a problem lies within a measurement or control system often reduces 
to testing various instruments to see if their output responses appropriately match their input 

For example, one way to test the flow transmitter in this system would be to subject it to a 
number of different pressures within its range (specified in the diagram as to 100 inches of water 
column differential) and seeing whether or not the current signal output by the transmitter was 
consistently proportional to the applied pressure (e.g. 4 mA at inches pressure, 20 mA at 100 
inches pressure, 12 mA at 50 inches pressure, etc.). 

Given the fact that a calibration error or malfunction in any one of these instruments can cause 
a problem for the control system as a whole, it is nice to know there is a way to determine which 
instrument is to blame and which instruments are not. This general principle holds true regardless 
of the instrument's type or technology. You can use the same input-versus-output test procedure to 
verify the proper operation of a pneumatic (3 to 15 PSI) level transmitter or an analog electronic 
(4 to 20 mA) flow transmitter or a digital (fieldbus) temperature transmitter alike. Each and every 
instrument has an input and an output, and there is always a predictable (and testable) correlation 
from one to the other. 

Another interesting detail seen on this loop diagram is the action of each instrument. You will 
notice a box and arrow (pointing either up or down) next to each instrument bubble. An "up" arrow 
(|) represents a direct-acting instrument: one whose output signal increases as the input stimulus 
increases. A "down" arrow (J,) represents a reverse-acting instrument: one whose output signal 
decreases as the input stimulus increases. All the instruments in this loop are direct-acting with the 
exception of the pressure differential transmitter PDT-42: 


Here, the "down" arrow tells us the transmitter will output a full-range signal (20 mA) when it 
senses zero differential pressure, and a 0% signal (4 mA) when sensing a full 200 PSI differential. 
While this calibration may seem confusing and unwarranted, it serves a definite purpose in this 
particular control system. Since the transmitter's current signal decreases as pressure increases, and 
the controller must be correspondingly configured, a decreasing current signal will be interpreted 
by the controller as a high differential pressure. If any wire connection fails in the 4-20 mA current 
loop for that transmitter, the resulting mA signal will be naturally "seen" by the controller as 
a pressure over-range condition. This is considered dangerous in a compressor system because it 
predicts a condition of surge. Thus, the controller will naturally take action to prevent surge by 
commanding the anti-surge control valve to open, because it "thinks" the compressor is about to 
surge. In other words, the transmitter is intentionally calibrated to be reverse-acting so that any 
break in the signal wiring will naturally bring the system to its safest condition. 



6.4 SAMA diagrams 

SAMA is an acronym standing for Scientific Apparatus Makers Association, referring to a unique 
form of diagram used primary in the power generation industry to document control strategies. 
These diagrams focus on the flow of information within a control system rather than on the process 
piping or instrument interconnections (wires, tubes, etc.). The general flow of a SAMA diagram is 
top-to-bottom, with the process sensing instrument (transmitter) located at the top and the final 
control element (valve or variable-speed motor) located at the bottom. No attempt is made to 
arrange symbols in a SAMA diagram to correlate with actual equipment layout: these diagrams are 
all about the algorithms used to make control decisions, and nothing more. 

A sample SAMA diagram appears here, showing a flow transmitter (FT) sending a process 
variable signal to a PID controller, which then sends a manipulated variable signal to a flow control 
valve (FCV): 

f FT J Flow transmitter 



PID controller 

FCV \ Flow control valve 

A cascaded control system, where the output of one controller acts as the setpoint for another 
controller to follow, appears in SAMA diagram form like this: 









1 >- 


P I D 

P I D 



f FT j Flow transmitter 

PID controller 

/ FCV \ Flow control valve 

In this case, the primary controller senses the level in a vessel, commanding the secondary (flow) 
controller to maintain the necessary amount of flow either in or out of the vessel as needed to 
maintain level at some setpoint. 

SAMA diagrams may show varying degrees of detail about the control strategies they document. 
For example, you may see the auto/manual controls represented as separate entities in a SAMA 
diagram, apart from the basic PID controller function. In the following example, we see a transfer 
block (T) and two manual adjustment blocks (A) providing a human operator the ability to 
separately adjust the controller's setpoint and output (manipulated) variables, and to transfer 
between automatic and manual modes: 



f FT J Flow transmitter 



PID controller 


/ FCV \ Flow control valve 

Rectangular blocks such as the A, P, I, and D shown in this diagram represent automatic 
functions. Diamond-shaped blocks such as the A and T blocks are manual functions which must 
be set by a human operator. Showing even more detail, the following SAMA diagram indicates the 
presence of setpoint tracking in the controller algorithm, a feature that forces the setpoint value to 
equal the process variable value any time the controller is in manual mode: 



f FT J Flow transmitter 




I D 

PID controller 


FCV \ Flow control valve 

Here we see a new type of line: dashed instead of solid. This too has meaning in the world 
of SAMA diagrams. Solid lines represent analog (continuously variable) signals such as process 
variable, setpoint, and manipulated variable. Dashed lines represent discrete (on/off) signal paths, 
in this case the auto/manual state of the controller commanding the PID algorithm to get its setpoint 
either from the operator's input (A) or from the process variable input (the flow transmitter: FT). 


6.5 Instrument and process equipment symbols 
6.5.1 Line types 

Process flow line 

Instrument supply 

or process connection 

(impulse line) 



—f A 

Pneumatic signal 


—H- H— 

Pneumatic signal 

(discrete -- on/off) 

—ft ft— 

Capillary tube 
— X X— 

Hydraulic signal 

b b- 

Electric signal 


-M +H— 

Electric signal 
(discrete -- on/off) 

— \— - \— 

-m — w^- 

Data link 
(system internal) 

Data link 
(between systems) 

Mechanical link 

Radio link 

/V /I/ 

Sonic or other wave 

6.5.2 Process/Instrument line connections 




Socket welded 



Heat/cool traced 

(direct) Welded 



6.5.3 Instrument bubbles 

Field mounted 

Main control panel 

Main control panel 

Auxiliary control panel Auxiliary control panel 
front-mounted rear-mounted 







6.5.4 Process valve types 



Globe valve Butterfly valve Ball valve 

^X^~ H/l- 


Gate valve Saunders valve Plug valve ^ a \\ va | ve 

-«- HOr- 

Pneumatic pinch valve 

Diaphragm valve Angle valve Three-way valve 

Pressure regulator 

Check valve 

Ball check valve 


Pressure relief 
or safety valve 




6.5.5 Valve actuator types 

Diaphragm Electric motor Solenoid 

/ M 

^^ HXH 


w/ hand jack 

Electric motor 
w/ hand jack 

Hand (manual) 





6.5.6 Valve failure mode 



6.5.7 Flow measurement devices (flowing left-to-right) 

Orifice plate Pitot tube Averging pitot tubes 




Turbine Target 

Positive displacement 




Ultrasonic Magnetic Wedge 


Flow nozzle Venturi 



6.5.8 Process equipment 
Pressure vessels 












O O 

Motor-driven fan 


axial compressor 





Conveyor belt 
O O 

heat exchanger 

Jacketed vessel 



6.5.9 SAMA diagram symbols 

PID controllers PI controller D-PI controller PD-I controller 



p i 


p i 

P I D 

Kj % 



Manual adjust Manual transfer 

A > <T 

Control valve 

/fcv \ 

control valve 



Control valve 
w/ positioner 



Transmitter Time delay summer Square root Characterizer 


E -f 



Instrumentation, Systems, and Automation Society Standards, 5.1-1984 (R1992), Instrumentation 
Symbols and Identification, Research Triangle Park, NC, 1984. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Software and Digital Networks, Third 
Edition, CRC Press, New York, NY, 2002. 


Chapter 7 

Discrete process measurement 

The word "discrete" means individual or distinct. In engineering, a "discrete" variable or 
measurement refers to a true-or-false condition. Thus, a discrete sensor is one that is only able 
to indicate whether the measured variable is above or below a specified setpoint. 

Discrete sensors typically take the form of switches, built to "trip" when the measured quantity 
either exceeds or falls below a specified value. These devices are less sophisticated than so-called 
continuous sensors capable of reporting an analog value, but they are quite useful in industry. 

Many different types of discrete sensors exist, detecting variables such as position, fluid pressure, 
material level, temperature, and fluid flow rate. The output of a discrete sensor is typically electrical 
in nature, whether it be an active voltage signal or just resistive continuity between two terminals 
on the device. 



7.1 "Normal" status of a switch 

Perhaps the most confusing aspect of discrete sensors is the definition of a sensor's normal status. 
Electrical switch contacts are typically classified as either normally-open or normally-closed, referring 
to the open or closed status of the contacts under "normal" conditions. But what exactly defines 
"normal" for a switch? The answer is not complex, but it is often misunderstood. 

The "normal" status for a switch is the status its electrical contacts are in under a condition of 
minimum physical stimulus. For a momentary-contact pushbutton switch, this would be the status 
of the switch contact when it is not being pressed. The "normal" status of any switch is the way 
it is drawn in an electrical schematic. For instance, the following diagram shows a normally-open 
pushbutton switch controlling a lamp on a 120 volt AC circuit (the "hot" and "neutral" poles of the 
AC power source labeled LI and L2, respectively): 




Normally-open contacts 

We can tell this switch is a normally-open (NO) switch because it is drawn in an open position. 
The lamp will energize only if someone presses the switch, holding its normally-open contacts in the 
"closed" position. Normally-open switch contacts are sometimes referred to in the electrical industry 
as form-A contacts. 

If we had used a normally-closed pushbutton switch instead, the behavior would be exactly 
opposite. The lamp would energize if the switch was left alone, but it would turn off if anyone 
pressed the switch. Normally-closed switch contacts are sometimes referred to in the electrical 
industry as form-B contacts. : 

Lj L 2 



• v — 'v 

Normally-closed contacts 

This seems rather simple, don't you think? What could possibly be confusing about the "normal" 
status of a switch? The confusion becomes evident, though, when you consider the case of a different 
kind of discrete sensor such as a flow switch. 

A flow switch is built to detect fluid flow through a pipe. In a schematic diagram, the switch 
symbol appears to be a toggle switch with a "flag" hanging below. The schematic diagram, of course, 
only shows the circuitry and not the pipe where the switch is physically mounted: 


A low coolant flow alarm circuit 
Lj L 2 

Flow switch lamp 

— f th 

This particular flow switch is used to trigger an alarm light if coolant flow through the pipe ever 
falls to a dangerously low level, and the contacts are normally-closed as evidenced by the closed 
status in the diagram. Here is where things get confusing: even though this switch is designated as 
"normally-closed," it will spend most of its lifetime being held in the open status by the presence of 
adequate coolant flow through the pipe. Only when the flow through the pipe slows down enough 
will this switch return to its "normal" status (remember, the condition of minimum stimulus?) and 
conduct electrical power to the lamp. In other words, the "normal" status of this switch (closed) is 
actually an abnormal status for the process it is sensing (low flow)! 

Students often wonder why process switch contacts are labeled according to this convention of 
"minimum stimulus" instead of according to the typical status of the process in which the switch 
is used. The answer to this question is that the manufacturer of the sensor has no idea whatsoever 
as to your intended use. The manufacturer of the switch does not know and does not care whether 
you intend to use their flow switch as a low-flow alarm or as a high-flow alarm. In other words, the 
manufacturer cannot predict what the typical status of your process will be, and so the definition 
of "normal" status for the switch must be founded on some common criterion unrelated to your 
particular application. That common criterion is the status of minimum stimulus: when the sensor 
is exposed to the least amount of stimulation from the process it senses. 

Here is a listing of "normal" definitions for various discrete sensor types: 

• Hand switch: no one pressing the switch 

• Limit switch: target not contacting the switch 

• Proximity switch: target far away 

• Pressure switch: low pressure (or even a vacuum) 

• Level switch: low level (empty) 

• Temperature switch: low temperature (cold) 

• Flow switch: low flow rate (fluid stopped) 

These are the conditions represented by the switch statuses shown in a schematic diagram. 
These may very well not be the statuses of the switches when they are exposed to typical operating 
conditions in the process. 



7.2 Hand switches 

A hand switch is exactly what the name implies: an electrical switch actuated by a person's hand 
motion. These may take the form of toggle, pushbutton, rotary, pull-chain, etc. A common form of 
industrial pushbutton switch looks something like this: 


NC terminal ^ 
NO terminal ^ 

Threaded neck 


^ NC terminal 
^ NO terminal 

The threaded neck inserts through a hole cut into a metal or plastic panel, with a matching nut 
to hold it in place. Thus, the button faces the human operator(s) while the switch contacts reside 
on the other side of the panel. 

When pressed, the downward motion of the actuator breaks the electrical bridge between the 
two NC contacts, forming a new bridge between the NO contacts: 

NC terminal @- 
NO terminal ^ 


Switch in the actuated 
(pressed) state 

^ NC terminal 
<§) NO terminal 

The schematic diagram symbol for this type of switch looks much like the real thing, with the 
normally-closed contact set on top and the normally-open contact set below: 


7.3 Limit switches 


Limit switch symbols 



A limit switch detects the physical motion of an object by direct contact with that object. 
An example of a limit switch is the switch detecting the open position of an automobile door, 
automatically energizing the cabin light when the door opens. 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A limit switch 
will be in its "normal" status when it is not in contact with anything (i.e. nothing touching the 
switch actuator mechanism). 

Limit switches find many uses in industry, particular in robotic control and CNC (Computer 
Numerical Control) machine tool systems. In many motion-control systems, the moving elements 
have "home" positions where the computer assigns a position value of zero. For example, the axis 
controls on a CNC machine tool such as a lathe or mill all return to their "home" positions upon 
start-up, so the computer can know with confidence the starting locations of each piece. These home 
positions are detected by means of limit switches. The computer commands each servo motor to 
travel fully in one direction until a limit switch on each axis trips. The position counter for each 
axis resets to zero as soon as the respective limit switch detects that the home position has been 

A typical limit switch design uses a roller-tipped lever to make contact with the moving part. 
Screw terminals on the switch body provide connection points with the NC and NO contacts inside 
the switch. Most limit switches of this design share a "common" terminal between the NC and NO 
contacts like this: 

Push lever down 
to actuate 

Roller tip 

Equivalent schematic 

This switch contact arrangement is sometimes referred to as a form-C contact set, since it 
incorporates both a form-A contact (normally-open) as well as a form-B contact (normally-closed). 




7.4 Proximity switches 

A proximity switch is one detecting the proximity (closeness) of some object. By definition, these 
switches are non-contact sensors, using magnetic, electric, or optical means to sense the proximity 
of objects. 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A proximity 
switch will be in its "normal" status when it is distant from any actuating object. 

Being non-contact in nature, proximity switches are often used instead of direct-contact limit 
switches for the same purpose of detecting the position of a machine part, with the advantage of 
never wearing out over time due to repeated physical contact. However, the greater complexity (and 
cost) of a proximity switch over a mechanical limit switch relegates their use to applications where 
lack of physical contact yields tangible benefits. 

Most proximity switches are active in design. That is, they incorporate a powered electronic 
circuit to sense the proximity of an object. Inductive proximity switches sense the presence of 
metallic objects through the use of a high-frequency magnetic field. Capacitive proximity switches 
sense the presence of non-metallic objects through the use of a high-frequency electric field. Optical 
switches detect the interruption of a light beam by an object. 

The schematic diagram symbol for a proximity switch with mechanical contacts is the same as 
for a mechanical limit switch, except the switch symbol is enclosed by a diamond shape, indicating 
a powered (active) device: 

Proximity switch symbols 

Normally-open Normally-closed 

(NO) (NC) 

Many proximity switches, though, do not provide "dry contact" outputs. Instead, their output 
elements are transistors configured either to source current or sink current. The terms "sourcing" 
and "sinking" are best understood by visualizing electric current in the direction of conventional 
flow rather than electron flow. The following schematic diagrams contrast the two modes of switch 
operation, using red arrows to show the direction of current (conventional flow notation) . In both 
examples, the load being driven by each proximity switch is a light- emit ting diode (LED): 



"Sinking" output 
proximity switch 

Current "sinks" down to 
ground through the switch 


"Sourcing" output 
proximity switch 


Switch "sources" current 
out to the load device 







This switch detects the passing of teeth on the chain sprocket, generating a slow square- wave 
electrical signal as the sprocket rotates. Such a switch may be used as a rotational speed sensor 



(sprocket speed proportional to signal frequency) or as a broken chain sensor (when sensing the 
rotation of the driven sprocket): 


RECTL'* ' : 









7.5 Pressure switches 

A pressure switch is one detecting the presence of fluid pressure. Pressure switches often use 
diaphragms or bellows as the pressure-sensing element, the motion of which actuates one or more 
switch contacts. 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A pressure 
switch will be in its "normal" status when it senses minimum pressure (e.g. no applied pressure, or 
in some cases a vacuum condition) . 

Pressure switch symbols 




The following photograph shows two pressure switches sensing the same fluid pressure as an 
electronic pressure transmitter (the device on the far left): 

J If the trip setting of a pressure switch is below atmospheric pressure, then it will be "actuated" at atmospheric 
pressure and in its "normal" status only when the pressure falls below that trip point (i.e. a vacuum). 



In this photograph, we see a pressure switch actuated by differential pressure (the difference in 
fluid pressure sensed between two ports): 

The electrical switch element is located underneath the blue cover, while the diaphragm pressure 
element is located within the grey metal housing. The net force exerted on the diaphragm by the 
two fluid pressures varies in magnitude and direction with the magnitude of those pressures. If 
the two fluid pressures are precisely equal, the diaphragm experiences no net force (zero differential 
pressure) . 



7.6 Level switches 

A level switch is one detecting the level of liquid or solid (granules or powder) in a vessel. Level 
switches often use floats as the level-sensing element, the motion of which actuates one or more 
switch contacts. 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A level switch 
will be in its "normal" status when it senses minimum level (e.g. an empty vessel). 

Level switch symbols 




Two water level switches appear in this photograph of an old boiler. The switches sense water 
level in the steam drum of the boiler. Both water level switches are manufactured by the Magnetrol 

The switch mechanism is a mercury tilt bulb, tilted by a magnet's attraction to a steel rod lifted 
into position by a float. The float directly senses liquid level, which positions the steel rod either 



closer to or further away from the magnet. If the rod comes close enough to the magnet, the mercury 
bottle will tilt and change the switch's electrical status. 

This level switch uses a metal tuning fork structure to detect the presence of a liquid or solid 
(powder or granules) in a vessel: 

An electronic circuit continuously excites the tuning fork, causing it to mechanically vibrate. 
When the prongs of the fork contact anything with substantial mass, the resonant frequency of 
the structure dramatically decreases. The circuit detects this change and indicates the presence of 
material contacting the fork. 


7.7 Temperature switches 

A temperature switch is one detecting the temperature of an object. Temperature switches often use 
bimetallic strips as the pressure-sensing element, the motion of which actuates one or more switch 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A temperature 
switch will be in its "normal" status when it senses minimum temperature (i.e. cold, in some cases 
a condition colder than ambient) 2 . 

Temperature switch symbols 


Normally-open Normally-closed 

(NO) (NC) 

The following photograph shows a temperature-actuated switch: 

2 If the trip setting of a temperature switch is below ambient temperature, then it will be "actuated" at ambient 
temperature and in its "normal" status only when the temperature falls below that trip point (i.e. colder than 





7.8 Flow switches 

A flow switch is one detecting the flow of some fluid through a pipe. Flow switches often use 
"paddles" as the flow-sensing element, the motion of which actuates one or more switch contacts. 

Recall that the "normal" status of a switch is the condition of minimum stimulus. A flow switch 
will be in its "normal" status when it senses minimum flow (i.e. no fluid moving through the pipe). 

Flow switch symbols 



A simple paddle placed in the midst of a fluid stream generates a mechanical force which may 
be used to actuate a switch mechanism, as shown in the following photograph: 


Chapter 8 

Analog electronic instrumentation 

8.1 4 to 20 mA analog current signals 

The most popular form of signal transmission used in modern industrial instrumentation systems 
(as of this writing) is the 4 to 20 milliamp DC standard. This is an analog signal standard, meaning 
that the electric current is used to proportionately represent measurements or command signals. 
Typically, a 4 milliamp current value represents 0% of scale, a 20 milliamp current value represents 
100% of scale, and any current value in between 4 and 20 milliamps represents a commensurate 
percentage in between 0% and 100%. 

For example, if we were to calibrate a 4-20 mA temperature transmitter for a measurement range 
of 50 to 250 degrees C, we could relate the current and measured temperature values on a graph like 





Measured 160 
temperature 150 









































DC current signal (mA) 

This is not unlike the pneumatic instrument signal standard or 3 to 15 pounds per square 
inch (PSI), where a varying air pressure signal represents some process measurement in an analog 
(proportional) fashion. 

DC current signals are also used in control systems to command the positioning of a final control 
element, such as a control valve or a variable-speed motor drive (VSD). In these cases, the milliamp 
value does not directly represent a process measurement, but rather how the degree to which the 
final control element influences the process. Typically (but not always!), 4 milliamps commands a 
closed (shut) control valve or a stopped motor, while 20 milliamps commands a wide-open valve or 
a motor running at full speed. 



Thus, most industrial control systems use at least two different 4-20 mA signals: one to represent 
the process variable (PV) and one to represent the command signal to the final control element (the 
"manipulated variable" or MV): 


4-20 mA 
PV signal^ 






4-20 mA 
v. MV signal 

nses 1 


V Influences 


Final control 

The Process 

The relationship between these two signals depends entirely on the response of the controller. 
There is no reason to ever expect the two current signals to be equal, for they represent entirely 
different things. In fact, if the controller is reverse-acting, it is entirely normal for the two current 
signals to be inversely related: as the PV signal increases going to a reverse-acting controller, the 
output signal will decrease. If the controller is placed into "manual" mode by a human operator, 
the output signal will have no automatic relation to the PV signal at all, instead being entirely 
determined by the operator's whim. 


8.2 Relating 4 to 20 mA signals to instrument variables 

Calculating the equivalent milliamp value for any given percentage of signal range is quite easy. 
Given the linear relationship between signal percentage and milliamps, the equation takes the form 
of the standard slope-intercept line equation y = mx + b. Here, y is the equivalent current in 
milliamps, x is the desired percentage of signal, m is the span of the 4-20 mA range (16 mA), and 
b is the offset value, or the "live zero" of 4 mA: 

current = (16 mA) I — ) + (4 mA) 


This equation form is identical to the one used to calculate pneumatic instrument signal pressures 
(the 3 to 15 PSI standard): 

pressure = (12 PSI) (^^^ j + (3 PSI) 

The same mathematical relationship holds for any linear measurement range. Given a percentage 
of range x, the measured variable is equal to: 

measured variable = (Span) I — ) + (LRV) 

v 100 /o / 

Some practical examples of calculations between milliamp current values and process variable 
values follow: 

8.2.1 Example calculation: controller output to valve 

An electronic loop controller outputs a signal of 8.55 mA to a direct-responding control valve (where 
4 mA is shut and 20 mA is wide open). How far open should the control valve be at this MV signal 

We must convert the milliamp signal value into a percentage of valve travel. This means 
determining the percentage value of the 8.55 mA signal on the 4-20 mA range. First, we need 
to manipulate the percentage-milliarnp formula to solve for percentage (x): 

(16 mA) ( — ) + (4 mA) = current 

V ; vioo%/ V ; 

(16 mA) I r- ) = current — (4 mA) 

V ; vioo%/ V ' 

x current — (4 mA) 

100% (16 mA) 

current — (4 mA) \ 

' l 100% 

(16 mA) 
Next, we plug in the 8.55 mA signal value and solve for x 


.55 mA- (4 mA)\ 

1 100% 

(16 mA) 

x = 28.4% 
Therefore, the control valve should be 28.4 % open when the MV signal is at a value of 8.55 mA. 

8.2.2 Example calculation: flow transmitter 

A flow transmitter is ranged to 350 gallons per minute, 4-20 mA output, direct-responding. 
Calculate the current signal value at a flow rate of 204 GPM. 

First, we convert the flow value of 204 GPM into a percentage of range. This is a simple matter 
of division, since the flow measurement range is zero-based: 

204 GPM 

3«)GPM= a583 = 58 - 3% 
Next, we take this percentage value and translate it into a milliamp value using the formula 
previously shown: 

(16 mA) ( — ) + (4 mA) = current 

V ; vioo%/ V ; 

(16 mA) [ ^45 I + (4 mA) = 13.3 mA 

V ; V ioo% J v ; 

Therefore, the transmitter should output a PV signal of 13.3 mA at a flow rate of 204 GPM. 

8.2.3 Example calculation: temperature transmitter 

A pneumatic temperature transmitter is ranged 50 to 140 degrees Fahrenheit and has a 3-15 PSI 
output signal. Calculate the pneumatic output pressure if the temperature is 79 degrees Fahrenheit. 

First, we convert the temperature value of 79 degrees into a percentage of range based on the 
knowledge of the temperature range span (140 degrees — 50 degrees = 90 degrees) and lower-range 
value (LRV = 50 degrees). We may do so by manipulating the general formula for any linear 
measurement to solve for x: 

measured variable = (Span) I — j + (LRV) 

measured variable — (LRV) = (Span) I — j 

measured variable — (LRV) x 

(Span) ~ 100% 

/ measured variable — (LRV) \ „ 
*=( (Span) ' )' 00% 





x = 32.2% 

Next, we take this percentage value and translate it into a pneumatic pressure value using the 
formula previously shown: 

(12 PSI) (^) + (3 PSI) = pressure 

(12 PSI) (^r) + (3 PSI) = 6.87 PSI 
Therefore, the transmitter should output a PV signal of 6.87 PSI at a temperature of 79° F. 

8.2.4 Example calculation: pH transmitter 

A pH transmitter has a calibrated range of 4 pH to 10 pH, with a 4-20 mA output signal. Calculate 
the pH sensed by the transmitter if its output signal is 11.3 mA. 

First, we must convert the milliamp value into a percentage. Following the same technique we 
used for the control valve problem: 

current — (4 mA) ' , 

1 100% = percent of range 

(16 mA) 
11.3 mA - (4 mA) 

100% = 0.456 = 45.6% 

(16 mA) 

Next, we take this percentage value and translate it into a pH value, given the transmitter's 
measurement span of 6 pH (10 pH — 4 pH)and offset of 4 pH: 

(10 PH) (j^r) + (4 pH) = pH value 

(10 PH) (^r) + (4 PH) = 8.56 P H 
Therefore, the transmitter's 11.3 mA output signal reflects a measured pH value of 8.56 pH. 

8.2.5 Example calculation: reverse-acting I/P transducer signal 

A current-to-pressure transducer is used to convert a 4-20 mA electronic signal into a 3-15 PSI 
pneumatic signal. This particular transducer is configured for reverse action instead of direct, 
meaning that its pressure output at 4 mA should be 15 PSI and its pressure output at 20 mA should 
be 3 PSI. Calculate the necessary current signal value to produce an output pressure of 12.7 PSI. 

Reverse-acting instruments are still linear, and therefore still follow the slope-intercept line 
formula y = mx + b. The only differences are a negative slope and a different intercept value. 


Instead of y = 16a; + 4 as is the case for direct-acting instruments, this reverse-acting instrument 
follows the linear equation y = — 16x + 20: 

(-16 mA) | X n _ ) + (20 mA) = current 

V ; vioo%/ V ' 

First, we need to to convert the pressure signal value of 12.7 PSI into a percentage of 3-15 PSI 
range. We will manipulate the percentage-pressure formula to solve for x: 

(12 PSI) ( T ^) + (3 PSI) = pressure 

(12 PSI) ( T ^) = pressure - (3 PSI) 
x pressure — (3 PSI) 

100% (12 PSI) 

pressure — (3 PSI) \ 
(12 PSI) J 


Next, we plug in the 12.7 PSI signal value and solve for x: 

12.7 PSI -(3 PSI) \ 
(12PSI) J 

x = 80.8% 

This tells us that 12.7 PSI represents 80.8 % of the 3-15 PSI signal range. Plugging this percentage 
value into our modified (negative-slope) percentage-current formula will tell us how much current is 
necessary to generate this 12.7 PSI pneumatic output: 

(-16 mA) ( X — J + (20 mA) = current 
v 100%/ 

(-16 mA) [ ^Sr J + ( 20 mA ) = 7m mA 
\ 100% / 

Therefore, a current signal of 7.07 mA is necessary to drive the output of this reverse-acting I/P 
transducer to a pressure of 12.7 PSI. 

8.2.6 Graphical interpretation of signal ranges 

A helpful illustration for students in understanding analog signal ranges is to consider the signal 
range to be expressed as a length on a number line. For example, the common 4-20 mA analog 
current signal range would appear as such: 

4 mA 8 mA 12 mA 16 mA 20 mA 

I 1 1 1 1 

0% 25% 50% 75% 100% 



If one were to ask the percentage corresponding to a 14.4 mA signal on a 4-20 mA range, it 
would be as simple as determining the length of a line segment stretching from the 4 mA mark to 
the 14.4 mA mark: 

14.4 mA 
4 mA 8 mA 12 mA u 16 mA 20 mA 





1 00% 

10.4 mA length 

16 mA span 

As a percentage, this thick line is 10.4 mA long (the distance between 14.4 mA and 4 mA) over 
total (possible) length of 16 mA (the total span between 20 mA and 4 mA). Thus: 


/14.4 mA- 4 mA\ 
l 20 mA - 4 mA J 


Percentage = 65% 

This same "number line" approach may be used to visualize any conversion from one analog scale 
to another. Consider the case of an electronic pressure transmitter calibrated to a pressure range of 
-5 to +25 PSI, having an (obsolete) current signal output range of 10 to 50 mA. The appropriate 
current signal value for an applied pressure of +12 PSI would be represented on the number line as 

+12 PSI 
-5 PSI +2.5 PSI +10 Psjl +17.5 PSI +25 PSI 

10 mA 20 mA 30 mA 40 mA 

50 mA 


17 PSI length 
? mA length 

17 PSI ?mA 

30 PSI 40 mA 

30 PSI span 
40 mA span 

Finding the "length" of this line segment in units of milliamps is as simple as setting up a 
proportion between the length of the line in units of PSI over the total (span) in PSI, to the length 
of the line in units of mA over the total (span) in mA: 


17 PSI ? mA 

30 PSI 40 mA 
Solving for the unknown (?) current by cross-multiplication and division yields a value of 22.67 
mA. Of course, this value of 22.67 mA only tells us the length of the line segment on the number 
line; it does not directly tell us the current signal value. To find that, we must add the "live zero" 
offset of 10 mA, for a final result of 32.67 mA. 

+12 PSI 
-5 PSI +2.5 PSI +10 Psll +17.5 PSI +25 PSI 

\ 1 r— 1 1 

10 mA 20 mA 30 mfii 40 mA 50 mA 


32.67 mA 

Thus, an applied pressure of +12 PSI to this transmitter should result in a 32.67 mA output 


8.3 Controller output current loops 

The simplest form of 4-20 mA current loop is the type used to represent the output of a process 
controller, sending a command signal to a final control element. Here, the controller both supplies 
the electrical power and regulates the DC current to the final control element, which acts as an 
electrical load. To illustrate, consider the example of a controller sending a 4-20 mA signal to an 
I/P (current-to-pressure) signal converter, which then pneumatically drives a control valve: 










2-wire cable 

20 PSI 
air supply 

Control valve 


(Current-to-Pressure converter) 

This particular controller has two digital displays, one for process variable (PV) and one for 
setpoint (SP), with a bargraph for displaying the output value (Out). One pushbutton provides 
the operator with a way to switch between Automatic and Manual modes (A/M), while two other 
pushbuttons provide means to decrement and increment either the setpoint value (in Automatic 
mode) or the Output value (in Manual mode). 

Inside the controller, a dependent current source provides the 4-20 mA DC current signal to 
the I/P transducer. Like all current sources, its purpose is to maintain current in the "loop" 
circuit regardless of circuit resistance or any external voltage sources. Unlike a constant current 
source, a "dependent" current source (represented by a diamond shape instead of a circle shape) 
varies its current value according to the dictates of some external stimulus. In this case, either the 
mathematical function of the controller (Automatic mode) or the arbitrary setting of the human 
operator (Manual mode) tells the current source how much DC current it should maintain in the 

For example, if the operator happened to switch the controller into Manual mode and set the 
output value at 50%, the proper amount of DC current for this signal percentage would be 12 mA 
(exactly half-way between 4 mA and 20 mA). If everything is working properly, the current in the 
"loop" circuit to the I/P transducer should remain exactly at 12 mA regardless of slight changes 
in wire resistance, I/P coil resistance, or anything else: the current source inside the controller will 
"fight" as hard as it has to in order to maintain this set amount of current. This current, as it flows 



through the wire coil of the I/P transducer mechanism, creates a magnetic field inside the I/P to 
actuate the pneumatic mechanism and produce a 9 PSI pressure signal output to the control valve 
(9 PSI being exactly half-way between 3 PSI and 15 PSI in the 3-15 PSI signal standard range). 
This should move the control valve to the half-way position. 

The details of the controller's internal current source are not terribly important. Usually, it takes 
the form of an operational amplifier circuit driven by the voltage output of a DAC (Digital-to- Analog 
Converter). The DAC converts a binary number (either from the controller's automatic calculations, 
or from the human operator's manual setting) into a small DC voltage, which then commands the 
op-amp circuit to regulate output current at a proportional value. 

The scenario is much the same if we replace the I/P and control valve with a variable-speed 
motor drive. From the controller's perspective, the only difference it sees is a resistive load instead 
of an inductive load. The input resistance of the motor drive circuit converts the 4-20 mA signal 
into an analog voltage signal (typically 1-5 V, but not always). This voltage signal then constitutes 
a command to the rest of the drive circuitry, telling it to modulate the power going to the electric 
motor in order to drive it at the desired speed: 





A / 

Dependent /\ +( 
current (f 

To source of 
3-phase AC power 



8.4 4- wire ("self-powered") transmitter current loops 

DC electric current signals may also be used to communicate process measurement information from 
transmitters to controllers, indicators, recorders, alarms, and other input devices. The simplest form 
of 4-20 mA measurement loop is one where the transmitter has two terminals for the 4-20 mA signal 
wires to connect, and two more terminals where a power source connects. These transmitters are 
called "4-wire" or self-powered. The current signal from the transmitter connects to the process 
variable input terminals of the controller to complete the loop: 



4-wire transmitter 

2-wire cable 









Typically, process controllers are not equipped to directly accept milliamp input signals, but 
rather voltage signals. For this reason we must connect a precision resistor across the input terminals 
to convert the 4-20 mA signal into a standardized analog voltage signal that the controller can 
understand. A voltage signal range of 1 to 5 volts is standard, although some models of controller 
use different voltage ranges and therefore require different precision resistor values. If the voltage 
range is 1-5 volts and the current range is 4-20 mA, the precision resistor value must be 250 ohms. 

Since this is a digital controller, the input voltage at the controller terminals is interpreted by 
an analog-to-digital converter (ADC) circuit, which converts the measured voltage into a digital 
number that the controller's microprocessor can work with. 

In some installations, the transmitter power is supplied through additional wires in the cable 
from a power source located in the same panel as the controller: 





4-wire transmitter 

4-wire cable 







A /„ 




The obvious disadvantage of this scheme is the requirement of two more conductors in the cable. 
More conductors means the cable will be larger-diameter and more expensive for a given length. 
Cables with more conductors will require larger electrical conduit to fit in to, and all field wiring 
panels will have to contain more terminal blocks to marshal the additional conductors. If no suitable 
electrical power source exists at the transmitter location, though, a 4-wire cable is necessary to service 
a 4-wire transmitter. 



8.5 2- wire ("loop-powered") transmitter current loops 

It is possible to convey electrical power and communicate analog information over the same two 
wires using 4 to 20 milliamps DC, if we design the transmitter to be loop-powered. A loop-powered 
transmitter connects to a process controller in the following manner: 



2-wire transmitter 

2-wire cable 









Here, the transmitter is not really a current source in the sense that a 4-wire transmitter is. 
Instead, a 2-wire transmitter's circuitry is designed to act as a current regulator, limiting current in 
the series loop to a value representing the process measurement, while relying on a remote source 
of power to motivate current to flow. Please note the direction of the arrow in the transmitter's 
dependent current source symbol, and how it relates to the voltage polarity marks. Refer back to the 
illustration of a 4-wire transmitter circuit for comparison. The current "source" in this loop-powered 
transmitter actually behaves as an electrical load, while the current source in the 4-wire transmitter 
functions as a true electrical source. 

A loop-powered transmitter gets its operating power from the minimum terminal voltage and 
current available at its two terminals. With the typical source voltage being 24 volts DC, and the 
maximum voltage dropped across the controller's 250 ohm resistor being 5 volts DC, the transmitter 
should always have at least 19 volts available at its terminals. Given the lower end of the 4-20 mA 
signal range, the transmitter should always have at least 4 mA of current to run on. Thus, the 
transmitter will always have a certain minimum amount of electrical power available on which to 
operate, while regulating current to signal the process measurement. 

Internally, the loop-powered transmitter circuitry looks something like this: 



Loop-powered 4-20 mA transmitter 






and scaling 





bias - 



feedback < 


Gnd R 




All sensing, scaling, and output conditioning circuitry inside the transmitter must be designed 
to run on less then 4 mA of DC current, and at a modest terminal voltage. In order to create loop 
currents exceeding 4 mA - as the transmitter must do in order to span the entire 4 to 20 milliamp 
signal range - the transmitter circuitry uses a transistor to shunt (bypass) extra current from one 
terminal to the other as needed to make the total current indicative of the process measurement. 
For example, if the transmitter's internal operating current is only 3.8 mA, and it must regulate loop 
current at a value of 16 mA to represent a condition of 75% process measurement, the transistor 
will bypass 12.2 mA of current. 

Early current-based industrial transmitters were not capable of operating on such low levels 
of electrical power, and so used a different current signal standard: 10 to 50 milliamps DC. 
Loop power supplies for these transmitters ranged upwards of 90 volts to provide enough power 
for the transmitter. Safety concerns made the 10-50 mA standard unsuitable for some industrial 
installations, and modern microelectronic circuitry with its reduced power consumption made the 
4-20 mA standard practical for nearly all types of process transmitters. 



8.6 Troubleshooting current loops 

Since the signal of interest is represented by an electric current in an instrumentation current "loop" 
circuit, the obvious tool to use for troubleshooting is a multimeter capable of accurately measuring 
DC milliamperes. Unfortunately, though, there is a major disadvantage to the use of a milliammeter: 
the circuit must be "broken" at some point to connect the meter in series with the current, and 
this means the current will fall to mA until the meter is connected (then fall to mA when 
the meter is removed from the circuit). Interrupting the current means interrupting the flow of 
information conveyed by that current, be it a process measurement or a command signal to a final 
control element. This will have adverse effects on a control system unless certain preparatory steps 
are taken. 

Before "breaking the loop" to connect your meter, one must first warn all appropriate personnel 
that the signal will be interrupted at least twice, falling to a value of -25% each time. If the signal to 
be interrupted is coming from a process transmitter to a controller, the controller should be placed 
in Manual mode so it will not cause an upset in the process (by moving the final control element 
in response to the sudden loss of PV signal). Also, process alarms should be temporarily disabled 
so that they do not cause panic. If this current signal also drives process shutdown alarms, these 
should be temporarily disabled so that nothing shuts down upon interruption of the signal. 

If the current signal to be interrupted is a command signal from a controller to a final control 
element, the final control element either needs to be manually overridden so as to hold a fixed setting 
while the signal varies, or it needs to be bypasses completely by some other device(s). If the final 
control element is a control valve, this typically takes the form of opening a bypass valve and closing 
at least one block valve: 

Control valve 



Since the manually-operated bypass valve now performs the job that the automatic control valve 
used to, a human operator must remain posted at the bypass valve to carefully throttle it and 
maintain control of the process. 

From this we see that the seemingly simple task of connecting a milliammeter in series with a 
4-20 mA current signal harbors certain risks and can be labor-intensive. Better ways must exist, 

One better way to measure a 4-20 mA signal without interrupting it is to do so magnetically, 
using a clamp-on milliammeter. Modern Hall-effect sensors are sensitive and accurate enough to 
now monitor the weak magnetic fields created by the passage of small DC currents in wires. Thus, 
a clamp-on milliammeter is very simple and non-intrusive to use. Not all technicians have access 
to these wonderful test instruments, though, and even if they do there are certain precautions one 
must take to ensure their indications will not be thrown into error by external magnetic fields. 

Another way to measure a 4-20 mA signal without interrupting it involves the use of a rectifying 
diode, originally installed in the loop circuit when it was commissioned. The diode may be placed 
anywhere in series within the loop in such a way that it will be forward-biased. During normal 
operation, the diode will drop approximately 0.7 volts, as is typical for any silicon rectifying diode 
when forward biased. The following schematic diagram shows such a diode installed in a 2-wire 
transmitter loop circuit: 

Transmitter \\ 

- + 


= 0.7 V 

77" supply 

If someone connects a milliammeter in parallel with this diode, however, the very low input 
resistance of the ammeters "shorts past" the diode and prevents any substantial voltage drop from 
forming across it. Without the necessary forward voltage drop, the diode effectively turns off and 
conducts mA, leaving the entire loop current to pass through the ammeter: 

All current goes through 
the milliammeter! 


_ Power 

When the milliammeter is disconnected, the requisite 0.7 volt drop appears to turn on the diode, 
and all loop current flows through the diode again. At no time is the loop current ever interrupted, 
which means a technician may take current measurements this way and never have to worry about 
generating false process variable indications, setting off alarms, or upsetting the process. 



Such a diode may be installed at the nearest junction box, between terminals on a terminal strip, 
or even incorporated into the transmitter itself. Some process transmitters have an extra pair of 
terminals labeled "Test" for this exact purpose. A diode is already installed in the transmitter, and 
these "test" terminals serve as points to connect the milliammeter across. 

A similar method for non-invasively measuring current in a 4-20 mA instrumentation circuit is to 
install a precision resistor in series. If the resistance value is precisely known, the technician merely 
needs to measure voltage across it with a voltmeter and use Ohm's Law to calculate current: 

Transmitter (\ 

V = IR 


If neither component (diode nor resistor) is pre-installed in the circuit, and if a Hall-effect (clamp- 
on) precision milliammeter is unavailable, a technician may still perform useful troubleshooting 
measurements using nothing but a DC voltmeter. Here, however, one must be careful of how to 
interpret these voltage measurements, for they may not directly correspond to the loop current as 
was the case with measurements taken in parallel with the precision resistor. 

Take for example this 4-20 mA loop where a controller sends a command signal to an I/P 










2-wire cable 


20 PSI 
air supply 

air tubing 

Control valve 


(Current-to-Pressure converter) I — I 



There is no standardized resistance value for I/P transducer coils, and so the amount of voltage 
dropped across the I/P terminals for any given amount of loop current will be unique for every 
different model of I/P. The Fisher model 567 I/P transducer built for 4-20 rnA signals has a nominal 
coil resistance of 176 ohms. Thus, we would expect to see a voltage drop of approximately 0.7 
volts at 4 mA and a drop of approximately 3.5 volts at 20 mA across the I/P terminals. Since the 
controller output terminals are directly in parallel with the I/P terminals, we would expect to see 
approximately the same voltage there as well (slightly greater due to wire resistance). The lack of 
known precision in the I/P coil resistance makes it difficult to tell exactly how much current is in 
the loop for any given voltage measurement we take with a voltmeter. However, if we do know the 
approximate coil resistance of the I/P, we can at least obtain an estimate of loop current, which is 
usually good enough for diagnostic purposes. 

If the I/P coil resistance is completely unknown, voltage measurements become useless for 
quantitative determination of loop current. Voltage measurements would be useful only for 
qualitatively determining loop continuity (i.e. whether there is a break in the wiring between the 
controller and I/P). 

Another example for consideration is this loop-powered 4-20 mA transmitter and controller 
circuit, where the controller supplies DC power for the loop: 


2-wire transmitter 

2-wire cable 

250 a 






A / 







It is very common to find controllers with their own built-in loop power supplies, due to the 
popularity of loop-powered (2-wire) 4-20 mA transmitters. If we know the transmitter requires a 
DC voltage source somewhere in the circuit to power it up, it makes sense to include one in the 
controller, right? 

The only voltage measurement that directly and accurately correlates to loop current is the 
voltage directly across the 250 ohm precision resistor. A loop current of 4 mA will yield a voltage 
drop of 1 volt, 12 mA will drop 3 volts, 20 mA will drop 5 volts, etc. 

A voltage measurement across the transmitter terminals will show us the difference in voltage 
between the 26 volt power supply and the voltage dropped across the 250 ohm resistor. In other 



words, the transmitter's terminal voltage is simply what is left over from the source voltage of 26 
volts after subtracting the resistor's voltage drop. This makes the transmitter terminal voltage 
inversely proportional to loop current: the transmitter sees approximately 25 volts at 4 mA loop 
current (0% signal) and approximately 21 volts at 20 mA loop current (100% signal). 

The use of the word "approximate" is very intentional here, for loop power supplies are usually 
non-regulated. In other words, the "26 volt" rating is approximate and subject to change! One of 
the advantages of the loop-powered transmitter circuit is that the source voltage is largely irrelevant, 
so long as it exceeds the minimum value necessary to ensure adequate power to the transmitter. If 
the source voltage drifts for any reason, it will have no impact on the measurement signal at all, 
because the transmitter is built as a current regulator, regulating current in the loop to whatever 
value represents the process measurement, regardless of slight changes in loop source voltage, wire 
resistance, etc. This rejection of power supply voltage changes means that the loop power supply 
need not be regulated, and so in practice it rarely is. 

This brings us to a common problem in loop-powered 4-20 mA transmitter circuits: maintaining 
sufficient operating voltage at the transmitter terminals. Recall that a loop-powered transmitter 
relies on the voltage dropped across its terminals (combined with a current of less than 4 mA) to 
power its internal workings. This means the terminal voltage must not be allowed to dip below a 
certain minimum value, or else the transmitter will not have enough electrical power to continue its 
normal operation. This makes it possible to "starve" the transmitter of voltage if the loop power 
supply voltage is insufficient, and/or if the loop resistance is excessive. 

To illustrate how this can be a problem, consider the following 4-20 mA measurement loop, where 
the controller supplies only 20 volts DC to power the loop, and an indicator is included in the circuit 
to provide operators with field-located indication of the transmitter's measurement: 



2-wire transmitter 

2-wire cable 

250 Q. 






A / 



20 VDC 




The indicator contains its own 250 ohm resistor to provide a 1-5 volt signal for the meter 
mechanism to sense. This means the total loop resistance is now 500 ohms (plus any wire resistance) . 
At full current (20 mA), this total resistance will drop (at least) 10 volts, leaving 10 volts or less 


at the transmitter terminals to power the transmitter's internal workings. 10 volts may not be 
enough for the transmitter to successfully operate, though. The Rosemount model 3051 pressure 
transmitter, for example, requires a minimum of 10.5 volts at the terminals to operate. 

However, the transmitter will operate just fine at lower loop current levels. When the loop 
current is only 4 mA, for example, the combined voltage drop across the two 250 ohm resistors will 
be only 2 volts, leaving about 18 volts at the transmitter terminals: more than enough for practically 
any model of 4-20 mA loop-powered transmitter to successfully operate. Thus, the problem of 
insufficient supply voltage only manifests itself when the process measurement nears 100% of range. 
This could be a difficult problem to diagnose, since it appears only during certain process conditions. 
A technician looking only for wiring faults (loose connections, corroded terminals, etc.) would never 
find the problem. 

When a loop-powered transmitter is starved of voltage, its behavior becomes erratic. This is 
especially true of "smart" transmitters with built-in microprocessor circuitry. If the terminal voltage 
dips below the required minimum, the microprocessor circuit shuts down. When the circuit shuts 
down, the current draw decreases accordingly. This causes the terminal voltage to rise again, at 
which point the microprocessor has enough voltage to start up. As the microprocessor "boots" back 
up again, it increases loop current to reflect the near-100% process measurement. This causes the 
terminal voltage to sag, which subsequently causes the microprocessor to shut down again. The 
result is a slow on/off cycling of the transmitter's current, which makes the process controller think 
the process variable is surging wildly. The problem disappears, though, as soon as the process 
measurement decreases enough that the transmitter is allowed enough terminal voltage to operate 


Chapter 9 

Pneumatic instrumentation 

While electricity is commonly used as a medium for transferring energy across long distances, it is 
also used in instrumentation to transfer information. A simple 4-20 mA current "loop" uses direct 
current to represent a process measurement in percentage of span, such as in this example: 


Pressure transmitter 

Applied pressure 




'V N 


i > 24 VDC 

The transmitter senses an applied fluid pressure from the process being measured, regulates 
electric current in the series circuit according to its calibration (4 mA = no pressure ; 20 mA = 
full pressure), and the indicator (ammeter) registers this measurement on a scale calibrated to read 
in pressure units. If the calibrated range of the pressure transmitter is to 250 PSI, then the 
indicator's scale will be labeled to read from to 250 PSI as well. No human operator reading that 
scale need worry about how the measurement gets from the process to the indicator - the 4-20 mA 
signal medium is transparent to the end- user as it should be. 

Air pressure may be used as an alternative signaling medium to electricity. Imagine a pressure 
transmitter designed to output a variable air pressure according to its calibration rather than a 
variable electric current. Such a transmitter would have to be supplied with a source of constant- 
pressure compressed air instead of an electric voltage, and the resulting output signal would be 
conveyed to the indicator via tubing instead of wires: 



Pressure transmitter 

20 PSI tube , , 
air. 4u.\ tube 

Applied pressure 

supply / ,°^p"' 


The indicator in this case would be a special pressure gauge, calibrated to read in units of 
process pressure although actuated by the pressure of clean compressed air from the transmitter 
instead of directly by process fluid. The most common range of air pressure for industrial pneumatic 
instruments is 3 to 15 PSI. An output pressure of 3 PSI represents the low end of the process 
measurement scale and an output pressure of 15 PSI represents the high end of the measurement 
scale. Applied to the previous example of a transmitter calibrated to a range of to 250 PSI, 
a lack of process pressure would result in the transmitter outputting a 3 PSI air signal and full 
process pressure would result in an air signal of 15 PSI. The face of this special "receiver" gauge 
would be labeled from to 250 PSI, while the actual mechanism would operate on the 3 to 15 PSI 
range output by the transmitter. Just like the 4-20 mA loop, the end-user need not know how the 
information gets transmitted from the process to the indicator. The 3-15 PSI signal medium is once 
again transparent to the operator. 

Pneumatic temperature, flow, and level control systems have all been manufactured to use the 
same principle of 3-15 PSI air pressure signaling. In each case, the transmitter and controller are both 
supplied clean compressed air at some nominal pressure (20 to 25 PSI, usually) and the instrument 
signals travel via tubing. The following illustrations show what some of these applications look like: 


Biodiesel "wash column" temperature control 

Wash water 

out r 

feed in < 


biodiesel out 

Foxboro model 12A 
temperature transmitter 

Instrument air 


Foxboro model 43 AP 

Out [_ 

supply (20 PSI) 

Spent wash 
water out 


Flow controller 

Flow transmitter 


20 PSI 1 
supply ' 



Flow control system 

Flow control valve 

Orifice plate 



Two-element boiler steam drum level control 

// lubing // 

Square-root extractor 

/m w\ 


20 PSI supply 

Level controller 

-H- H- 

Flow transmitter 
FT ) Steam 



Instruments functioning on compressed air, and process measurement signals transmitted as air 
pressures through long runs of metal tubing, was the norm for industrial instrumentation prior to 
the advent of reliable electronics. In honor of this paradigm, instrument technicians were often 
referred to as instrument mechanics, for these air-powered devices were mechanically complex and 
in frequent need of adjustment to maintain high accuracy. 

Pneumatic instruments still find wide application in industry, although it is increasingly rare to 
encounter completely pneumatic control loops. One of the most common applications for pneumatic 
control system components is control valve actuation, where pneumatic technology still dominates. 
Not only is compressed air used to create the actuation force in many control valve mechanisms, it is 
still often the signal medium employed to command the valve's position. Quite often this pneumatic 
signal originates from a device called an I/P transducer, or current-to-pressure converter, taking a 
4-20 mA control signal from the output of an electronic controller and translating that information 
as a pneumatic 3-15 PSI signal to the control valve's positioner or actuator. 



9.1 Pneumatic sensing elements 

Most pneumatic instruments use a simple but highly sensitive mechanism for converting mechanical 
motion into variable air pressure: the baffte-and-nozzle assembly (sometimes referred to as a flapper- 
and-nozzle assembly). A baffle is nothing more than a flat object obstructing the flow of air out of 
a small nozzle by close proximity: 

Pressure gauge 

From compressed 
air supply 
(20 PSI) 





Backpressure at 
nozzle (PSI) 

"i — i — i — i — i — i — i — i — i — i 
123456789 10 

Clearance, mils (thousandths of an inch) 

The physical distance between the baffle and the nozzle alters the resistance of air flow through 
the nozzle. This in turn affects the air pressure built up inside the nozzle (called the nozzle 
backpressure). Like a voltage divider circuit formed by one fixed resistor and one variable resistor, 
the bafHe/nozzle mechanism "divides" the pneumatic source pressure to a lower value based on the 
ratio of restrictiveness between the nozzle and the fixed orifice. 

This crude assemblage is surprisingly sensitive, as shown by the graph. With a small enough 
orifice, just a few thousandths of an inch of motion is enough to drive the pneumatic output between 
its saturation limits. Pneumatic transmitters typically employ a small sheet-metal lever as the 
baffle. The slightest motion imparted to this baffle by changes in the process variable (pressure, 
temperature, flow, level, etc.) detected by some sensing element will cause the air pressure to 



change in response. 

The principle behind the operation of a baffle/nozzle mechanism is often used directly in quality- 
control work, checking for proper dimensioning of machined metal parts. Take for instance this 
shaft diameter checker, using air to determine whether or not a machined shaft inserted by a human 
operator is of the proper diameter after being manufactured on an assembly line: 

From compressed 
air supply — ► 
(20 PSI) 


Test jig 

<>, J) P ressure 9 au 9 e 

- Clearance 


Machined metal shaft 

If the shaft diameter is too small, there will be excessive clearance between the shaft and the 
inside diameter of the test jig, causing less air pressure to register on the gauge. Conversely, if 
the shaft diameter is too large, the clearance will be less and the gauge will register a greater air 
pressure because the flow of air will be obstructed by the reduced clearance. The exact pressure is 
of no particular consequence to the quality-control operator reading the gauge. What does matter 
is that the pressure falls within an acceptable range, reflecting proper manufacturing tolerances for 
the shaft. In fact, just like the 3-15 PSI "receiver gauges" used as pneumatic instrument indicators, 
the face of this pressure gauge might very well lack pressure units (such as kPa or PSI), but rather 
be labeled with a colored band showing acceptable limits of mechanical fit: 

This is another example of the analogue nature of pneumatic pressure signals: the pressure 
registered by this gauge represents a completely different variable, in this case the mechanical fit of 
the shaft to the test jig. 


Although it is possible to construct a pneumatic instrument consisting only of a baffle/nozzle 
mechanism, this is rarely done. Usually the baffle/nozzle mechanism is but one of several components 
that comprise a "balancing" mechanism in a pneumatic instrument. It is this concept of self- 
balancing that we will study next. 


9.2 Self-balancing pneumatic instrument principles 

A great many precision instruments use the principle of balance to measure some quantity. Perhaps 
the simplest example of a balance-based instrument is the common balance-beam scale used to 
measure mass in a laboratory: 

/ I 

A specimen of unknown mass is placed in one pan of the scale, and precise weights are placed in 
the other pan until the scale achieves a condition of balance. When balance is achieved, the mass of 
the sample is known to be equal to the sum total of mass in the other pan. An interesting detail to 
note about the scale itself is that it has no need of routine calibration. There is nothing to "drift" 
out of spec which would cause the scale to read inaccurately. In fact, the scale itself doesn't even 
have a gauge to register the mass of the specimen: all it has is a single mark indicating a condition 
of balance. To express this more precisely, the balance beam scale is actually a differential mass 
comparison device, and it only needs to be accurate at a single point: zero. In other words, it only 
has to be correct when it tells you there is zero difference in mass between the specimen and the 
standard masses piled on the other pan. 

The elegance of this mechanism allows it to be quite accurate. The only real limitation to 
accuracy is the certainty to which we know the masses of the balancing weights. 

Imagine being tasked with the challenge of automating this laboratory scale. Suppose we grew 
weary of having to pay a lab technician to place standard weights on the scale to balance it for every 
new measurement, and we decided to find a way for the scale to balance itself. Where would we 
start? Well, we would need some sort of mechanism to tell when the scale was out of balance, and 
another mechanism to change weight on the other pan whenever an out-of-balance condition was 

The baffle/nozzle mechanism previously discussed would suffice quite well as a detection 
mechanism. Simply attach a baffle to the end of the pointer on the scale, and attach a nozzle 
adjacent to the baffle at the "zero" position (where the pointer should come to a rest at balance): 



/ I 

Tube Air supply 

Now we have a highly sensitive means of indicating when the scale is balanced, but we still have 
not yet achieved full automation. The scale cannot balance itself, at least not yet. 

What if, instead of using precise, machined, brass weights placed on the other pan to counter the 
mass of the specimen, we used a pneumatically-actuated force generator operated by the backpressure 
of the nozzle? An example of such a "force generator" is a bellows: a device made of thin sheet metal 
with circular corrugations in it, so that it looks like the bellows fabric on an accordion. Pneumatic 
pressure applied to the interior of the bellows causes it to elongate. If the metal of the bellows is 
flexible enough so it does not naturally restrain the motion of expansion, the force generated by the 
expansion of the bellows will almost exactly equal that predicted by the force-pressure-area equation: 



Force = Pressure x Area 

F = PA 


Applied pressure 

If the bellows' expansion is externally restrained so it does not stretch appreciably - and therefore 
the metal never gets the opportunity to act as a restraining spring - the force exerted by the bellows 
on that restraining object will exactly equal the pneumatic pressure multiplied by the cross-sectional 
area of the bellows' end. 



Applying this to the problem of the self-balancing laboratory scale, imagine fixing a bellows to 
the frame of the scale so that it presses downward on the pan where the brass weights normally go, 
then connecting the bellows to the nozzle backpressure: 

Air supply 

Now the scale will self-balance. When mass is added to the left-hand pan, the pointer (baffle) will 
move ever so slightly toward the nozzle until enough backpressure builds up behind the nozzle to make 
the bellows exert the proper amount of balancing force and bring the pointer back (very close) to 
its original balanced condition. This balancing action is entirely automatic: the nozzle backpressure 
adjusts to whatever it needs to be in order to keep the pointer at the balanced position, applying or 
venting pressure to the bellows as needed to keep the system in a condition of equilibrium. What 
we have created is a negative feedback system, where the output of the system (nozzle backpressure) 
continuously adjusts to match and balance the input (the applied mass). 

This is all well and good, but how does this help us determine the mass of the specimen in the 
left-hand pan? What good is this self-balancing scale if we cannot read the balancing force? All we 
have achieved so far is to make the scale self-balancing. The next step is making the balancing force 
readable to a human operator. 

Before we add the final piece to this automated scale, it is worthwhile to reflect on what has been 
done so far. By adding the baffle/nozzle and bellows mechanisms to the scale, we have abolished 
the need for brass weights and instead have substituted air pressure. In effect, the scale translates 
the specimen's mass into a proportional, analogue, air pressure. What we really need is a way to 
now translate that air pressure into a human-readable indication of mass. 

The solution is simple: add the pressure gauge back to the system. The gauge will register air 
pressure, but this time the air pressure will be proportionately equivalent to specimen mass. In 
honor of this proportionality, we may label the face of the pressure gauge in units of grams (mass) 
instead of PSI or kPa (pressure): 




Although it may seem as though we are done with the task of fully automating the laboratory 
scale, we can go a step further. Building this pneumatic negative-feedback balancing system provides 
us with a capability the old manually-operated scale never had: remote indication. There is no reason 
why the indicating gauge must be located near the scale. Nothing prevents us from locating the 
receiver gauge some distance from the scale, and using long lengths of tubing to connect the two: 


A long ways away 

Air supply 

Nozzle I Orifice 

By equipping the scale with a pneumatic self-balancing apparatus, we have turned it into a 
pneumatic mass transmitter, capable of relaying the mass measurement in pneumatic, analog form 
to an indicating gauge far away. This is the basic force-balance principle used in most pneumatic 
industrial transmitters to convert some process measurement into a 3-15 PSI pneumatic signal. 


9.3 Pilot valves and pneumatic amplifying relays 

Self-balancing mechanisms such as the fictitious pneumatic laboratory scale in the previous section 
are most accurate when the imbalance detection mechanism is most sensitive. In other words, the 
more aggressively the baffle/nozzle mechanism responds to slight out-of-balance conditions, the more 
precise will be the relationship between measured variable (mass) and output signal (air pressure to 
the gauge). 

A plain baffle/nozzle mechanism may be made extremely sensitive by reducing the size of the 
orifice. However, a problem caused by decreasing orifice size is a corresponding decrease in the 
nozzle's ability to provide increasing backpressure to fill a bellows of significant volume. In other 
words, a smaller orifice will result in greater sensitivity to baffle motion, but it also limits the air flow 
rate available to fill the bellows, which makes the system slower to respond. Another disadvantage of 
smaller orifices is that they become more susceptible to plugging due to impurities in the compressed 

An alternative technique to making the baffle/nozzle mechanism more sensitive is to amplify its 
output pressure using some other pneumatic device. This is analogous to increasing the sensitivity 
of a voltage-generating electrical detector by passing its output voltage signal through an electronic 
amplifier. Small changes in detector output become bigger changes in amplifier output which then 
causes our self-balancing system to be even more precise. 

What we need, then, is a pneumatic amplifier: a mechanism to amplify small changes in air 
pressure and convert them into larger changes in air pressure. In essence, we need to find a pneumatic 
equivalent of the electronic transistor: a device that lets one signal control another. 

First, let us analyze the following pneumatic mechanism and its electrical analogue (as shown on 
the right): 



Pneumatic mechanism 

air supply 

Equivalent electrical circuit 

Output pressure 




Control rod 
moves up 
and down 

As the control rod is moved up and down by an outside force, the distance between the plug 
and the seat changes. This changes the amount of resistance experienced by the escaping air, 
thus causing the pressure gauge to register varying amounts of pressure. There is little functional 
difference between this mechanism and a baffle/nozzle mechanism. Both work on the principle of one 
variable restriction and one fixed restriction (the orifice) "dividing" the pressure of the compressed 
air source to some lesser value. 

The sensitivity of this pneumatic mechanism may be improved by extending the control rod and 
adding a second plug/seat assembly. The resulting mechanism, with dual plugs and seats, is known 
as a pneumatic pilot valve. An illustration of a pilot valve is shown here, along with its electrical 
analogue (on the right): 



Pneumatic pilot valve 


air supply 

Equivalent electrical circuit 

Output pressure 


v n 


Control rod 

moves up 

u and down 

As the control rod is moved up and down, both variable restrictions change in complementary 
fashion. As one restriction opens up, the other pinches shut. The combination of two restrictions 
changing in opposite direction results in a much more aggressive change in output pressure as 
registered by the gauge. 

A similar design of pilot valve reverses the directions of the two plugs and seats. The only 
operational difference between this pilot valve and the previous design is an inverse relationship 
between control rod motion and pressure: 



Pneumatic pilot valve 


air supply 

Equivalent electrical circuit 

Output pressure 

Control ___] 



Control rod 
moves up 
and down 

At this point, all we've managed to accomplish is build a better baffle/nozzle mechanism. We 
still do not yet have a pneumatic equivalent of an electronic transistor. To accomplish that, we must 
have some way of allowing an air pressure signal to control the motion of a pilot valve's control rod. 
This is possible with the addition of a diaphragm, as shown in this illustration: 



air supply 

Output pressure 


Input pressure 

The diaphragm is nothing more than a thin disk of sheet metal, upon which an incoming air 
pressure signal presses. Force on the diaphragm is a simple function of signal pressure (P) and 
diaphragm area (A), as described by the standard force-pressure-area equation: 


If the diaphragm is taut, the elasticity of the metal allows it to also function as a spring. This 
allows the force to translate into displacement (motion), forming a definite relationship between 
applied air pressure and control rod position. Thus, the applied air pressure input will exert control 
over the output pressure. The addition of an actuating mechanism to the pilot valve turns it into a 
pneumatic relay, which is the pneumatic equivalent of the electronic transistor we were looking for. 

It is easy to see how the input air signal exerts control over the output air signal in these two 



air supply 

(High) output pressure 


air supply 

(High) input pressure 

(Low) output pressure 


(Low) input pressure 

Since there is a direct relationship between input pressure and output pressure in this pneumatic 
relay, we classify it as a direct-acting relay. If we were to add an actuating diaphragm to the first 
pilot valve design, we would have a reverse-acting relay as shown here: 

air supply 

Output pressure 


Input pressure 

The gain (A) of any pneumatic relay is defined just the same as the gain of any electronic 
amplifier circuit, the ratio of output change to input change: 




For example, if an input pressure change of A2 PSI results in an output pressure change of A12 
PSI, the gain of the pneumatic relay is 6. 

The Foxboro corporation used a very sensitive amplifying relay in many of their pneumatic 





leaf spring 


. Output 

■ — *~ signal 

, ,, Pneumatic 

(vent) . 

-i n amplifying 



The motion of the diaphragm actuated a pair of valves: one with a cone-shaped plug and the 
other with a metal ball for a plug. The ball-plug allowed supply air to go to the output port, while 
the cone-shaped "stem valve" plug vented excess air pressure to the vent port. 

The Fisher corporation used a different style of amplifying relay in some of their pneumatic 










The gain of this Fisher relay was much less than that of the Foxboro relay, since output pressure in 
the Fisher relay was allowed to act against input pressure by exerting force on a sizable diaphragm. 
The movable vent seat in the Fisher relay made this design a "non-bleeding" type, meaning it 
possessed the ability to close both supply and vent valves at the same time, allowing it to hold an 
output air pressure between saturation limits without bleeding a substantial amount of compressed 
air to atmosphere through the vent. The Foxboro relay design, by contrast, was a "bleeding type," 
whose ball and stem valves could never close simultaneously, and thus would always bleed some 
compressed air to atmosphere so long as the output pressure remained somewhere between saturation 



9.4 Analogy to opamp circuits 

Self-balancing pneumatic instrument mechanisms are very similar to negative-feedback operational 
amplifier circuits, in that negative feedback is used to generate an output signal in precise proportion 
to an input signal. This section compares simple operational amplifier ("opamp") circuits with 
analogous pneumatic mechanisms for the purpose of illustrating how negative feedback works, and 
learning how to generally analyze pneumatic mechanisms. 

In the following illustration, we see an opamp with no feedback (open loop), next to a baffle/nozzle 
mechanism with no feedback (open loop): 

V ir 

Air supply 





For each system there is an input and an output. For the opamp, input and output are both 
electrical (voltage) signals: Vi n is the differential voltage between the two input terminals, and V ou t 
is the single-ended voltage measured between the output terminal and ground. For the baffle/nozzle, 
the input is the physical gap between the baffle and nozzle (xi n ) while the output is the backpressure 
indicated by the pressure gauge (P ou t)- 

Both systems have very large gains. Operational amplifier open-loop gains typically exceed 
200,000 (over 100 dB), and we have already seen how just a few thousandths of an inch of baffle 
motion is enough to drive the backpressure of a nozzle nearly to its limits (supply pressure and 
atmospheric pressure, respectively). 

Gain is always defined as the ratio between output and input for a system. Mathematically, it is 
the quotient of output change and input change, with "change" represented by the triangular Greek 
capital-letter delta: 



Normally, gain is a unitless ratio. We can easily see this for the opamp circuit, since both output 
and input are voltages, any unit of measurement for voltage would cancel in the quotient, leaving a 
unitless quantity. This is not so evident in the baffle/nozzle system, with the output represented in 
units of pressure and the input represented in units of distance. 

If we were to add a bellows to the baffle/nozzle mechanism, we would have a system that inputs 
and outputs fluid pressure, allowing us to more formally define the gain of the system as a unitless 
ratio of A p"' : 



Air supply 




The general effect of negative feedback is to decrease the gain of a system, and also make that 
system's response more linear over the operating range. This is not an easy concept to grasp, 
however, and so we will explore the effect of adding negative feedback in detail for both systems. 
The simplest expression of negative feedback is a condition of 100% negative feedback, where the 
whole strength of the output signal gets "fed back" to the amplification system in degenerative 
fashion. For an opamp, this simply means connecting the output terminal directly to the inverting 
input terminal: 


We call this "negative" or "degenerative" feedback because its effect is counteractive in nature. 
If the output voltage rises too high, the effect of feeding this signal to the inverting input will be to 
bring the output voltage back down again. Likewise, if the output voltage is too low, the inverting 
input will sense this and act to bring it back up again. Self- correction typifies the very nature of 
negative feedback. 

Having connected the inverting input directly to the output of the opamp leaves us with the 
noninverting terminal as the sole remaining input. Thus, our input voltage signal is a ground- 
referenced voltage just like the output. The voltage gain of this circuit is unity (1), meaning that 
the output will assume whatever voltage level is present at the input, within the limits of the opamp's 
power supply. If we were to send a voltage signal of 5 volts to the noninverting terminal of this 
opamp circuit, it would output 5 volts, provided that the power supply exceeds 5 volts in potential 
from ground. 

Let's analyze exactly why this happens. First, we will start with the equation representing the 
open-loop output of an opamp, as a function of its differential input voltage: 

V m 


( V in(+) 




As stated before, the open-loop voltage gain of an opamp is typically very large (Aol = 200,000 
or more!). Connecting the opamp's output to the inverting input terminal simplifies the equation: 
V ou t may be substituted for Vi n i_\, and V^„( + ) simply becomes Vi„ since it is now the only remaining 
input. Reducing the equation to the two variables of V ou t and Vi n and a constant (Aol) allows us to 
solve for overall voltage gain (-jr 1 ) as a function of the opamp's internal voltage gain (Aol)- The 
following sequence of algebraic manipulations shows how this is done: 

V m 


(Vin ~ Vout) 

v m 


^OLV out 

AoLVout + V ., 


V out (Aol + 1) = A OL V m 

Overall gain 




Aol + 1 

If we assume an internal opamp gain of 200,000, the overall gain will be very nearly equal to 
unity (0.999995). Moreover, this near-unity gain will remain quite stable despite large changes in 
the opamp's internal (open-loop) gain. The following table shows the effect of major Aol changes 
on overall voltage gain (Ay): 

Internal gain 

A v 
Overall gain 











Note how an order of magnitude change 1 in Aol (from 100,000 to 1,000,000) results is a miniscule 
change in overall voltage gain (from 0.99999 to 0.999999). Negative feedback clearly has a stabilizing 
effect on the closed-loop gain of the opamp circuit, which is the primary reason it finds such wide 
application in engineered systems. It was this effect that led Harold Black in the late 1920's to apply 
negative feedback to the design of very stable telephone amplifier circuits. 

If we subject our negative feedback opamp circuit to a constant input voltage of exactly 5 volts, 
we may expand the table to show the effect of changing open-loop gain on the output voltage, and 
also the differential voltage appearing between the opamp's two input terminals: 

1 An "order of magnitude" is nothing more than a ten-fold change. Do you want to sound like you're really smart 
and impress those around you? Just start comparing ordinary differences in terms of orders of magnitude. "Hey dude, 
that last snowboarder's jump was an order of magnitude higher than the one before!" "Whoa, that's some big air . 
. ." Just don't make the mistake of using decibels in the same way ("Whoa dude, that last jump was at least 10 dB 
higher than the one before!" ) - you don't want people to think you're a nerd. 



Internal gain 

A v 
Overall gain 

v out 

Output voltage 

l / m(+) - Vin(-) 

Differential input voltage 





















With such extremely high open-loop voltage gains, it hardly requires any difference in voltage 
between the two input terminals to generate the necessary output voltage to balance the input. 
Thus, V ou t = Vi n for all practical purposes. 

One of the "simplifying assumptions" electronics technicians and engineers make when analyzing 
opamp circuits is that the differential input voltage in any negative feedback circuit is zero. As we 
see in the above table, this assumption is very nearly true. Following this assumption to its logical 
consequence allows us to predict the output voltage of any negative feedback opamp circuit quite 
simply. For example: 

V«=0|iV -( 
V, „ = 5 volts 

* V out = 5 volts 

If we simply assume there will be no difference of voltage between the two input terminals of the 
opamp with negative feedback in effect, we may conclude that the output voltage is exactly equal to 
the input voltage, since that is what must happen in order for the two input terminals to see equal 

Now let us apply similar techniques to the analysis of a pneumatic baffle/nozzle mechanism. 
Suppose we arrange a pair of identical bellows in opposition to one another on a force beam, so that 
any difference in force output by the two bellows will push the baffle either closer to the nozzle or 
further away from it: 



Air supply 





It should be clear that the left-hand bellows, which experiences the same pressure (Pout) a s the 
pressure gauge, introduces negative feedback into the system. If the output pressure happens to 
rise too high, the baffle will be pushed away from the nozzle by the force of the feedback bellows, 
causing backpressure to decrease and stabilize. Likewise, if the output pressure happens to go too 
low, the baffle will move closer to the nozzle and cause the backpressure to rise again. Once again 
we see the defining characteristic of negative feedback in action: its self-correcting nature works to 
counteract any change in output conditions. 

As we have seen already, the baffle/nozzle is exceptionally sensitive to motion. Only a few 
thousandths of an inch of motion is sufficient to saturate the nozzle backpressure at either extreme 
(supply air pressure or zero, depending on which direction the baffle moves). This is analogous to the 
differential inputs of an operational amplifier, which only need to see a few microvolts of potential 
difference to saturate the amplifier's output. 

Introducing negative feedback to the opamp led to a condition where the differential input voltage 
was held to (nearly) zero. In fact, this potential is so small that we safely considered it zero for 
the purpose of more easily analyzing the output response of the system. We may make the exact 
same "simplifying assumption" for the pneumatic mechanism: we will assume the baffle/nozzle gap 
remains constant in order to more easily determine the output pressure response to an input pressure. 

If we simply assume the baffle/nozzle gap cannot change with negative feedback in effect, we may 
conclude that the output pressure is exactly equal to the input pressure for the pneumatic system 
shown, since that is what must happen in order for the two pressures to generate exactly opposing 
forces so that the baffle will not move from its original position. 

The analytical technique of assuming perfect balance in a negative feedback system works just 
as well for more complicated systems. Consider the following opamp circuit: 



Here, negative feedback occurs through a voltage divider from the output terminal to the inverting 
input terminal, so that only one-half of the output voltage gets "fed back" degeneratively. If we 
follow our simplifying assumption that perfect balance (zero difference of voltage) will be achieved 
between the two opamp input terminals due to the balancing action of negative feedback, we are 
led to the conclusion that V ou t must be exactly twice the magnitude of Vi n . In other words, the 
output voltage must increase to twice the value of the input voltage in order for the divided feedback 
signal to exactly equal the input signal. Thus, feeding back half the output voltage yields an overall 
voltage gain of two. 

If we make the same (analogous) change to the pneumatic system, we see the same effect: 

Air supply 

Small bellows 
Area = 7, A 

Large bellows 
Area = A 


Here, the feedback bellows has been made smaller (exactly half the surface area of the input 
bellows). This results in half the amount of force applied to the force beam for the same amount 
of pressure. If we follow our simplifying assumption that perfect balance (zero baffle motion) will 
be achieved due to the balancing action of negative feedback, we are led to the conclusion that P ou t 
must be exactly twice the magnitude of P in . In other words, the output pressure must increase to 
twice the value of the input pressure in order for the divided feedback force to exactly equal the 
input force and prevent the baffle from moving. Thus, our pneumatic mechanism has a pressure 
gain of two, just like the opamp circuit with divided feedback. 



We could have achieved the same effect by moving the feedback bellows to a lower position on 
the force beam instead of changing its surface area: 

Air supply 



This arrangement effectively reduces the feedback force by placing the feedback bellows at a 
mechanical disadvantage to the input bellows. If the distance between the feedback bellows tip and 
the force beam pivot is exactly half the distance between the input bellows tip and the force beam 
pivot, the effective force ratio will be one-half. 

Pneumatic instruments built such that bellows' forces directly oppose one another in the same 
line of action to constrain the motion of a beam are known as "force balance" systems. Instruments 
built such that bellows' forces oppose one another through different lever lengths (such as in the 
last system) are technically known as "moment balance" systems, referencing the moment arm 
lengths through which the bellows' forces act to balance each other. However, one will often find 
that "moment balance" instruments are commonly referred to as "force balance" because the two 
principles are so similar. 

An entirely different classification of pneumatic instrument is known as motion balance. The 
same "simplifying assumption" of zero baffle/nozzle gap motion holds true for the analysis of these 
mechanisms as well: 

Air supply 




In this mechanism there is no fixed pivot for the beam. Instead, the beam hangs between the 
ends of two bellows units, affixed by pivoting links. As input pressure increases, the input bellows 



expands outward, attempting to push the beam closer to the nozzle. However, if we follow our 
assumption that negative feedback holds the nozzle gap constant, we see that the feedback bellows 
must expand the same amount, and thus (if it has the same area and spring characteristics as the 
input bellows) the output pressure must equal the input pressure: 

Air supply 


We call this a motion balance system instead of a force balance system because we see two 
motions canceling each other out to maintain a constant nozzle gap instead of two forces canceling 
each other out to maintain a constant nozzle gap. 

The gain of a motion-balance pneumatic instrument may be changed by altering the bellows-to- 
nozzle distance so that one of the two bellows has more effect than the other. For instance, this 
system has a gain of 2, since the feedback bellows must move twice as far as the input bellows in 
order to maintain a constant nozzle gap: 

Air supply 





Force-balance (and moment-balance) instruments are generally considered more accurate than 
motion-balance instruments because motion-balance instruments rely on the pressure elements 
(bellows, diaphragms, or bourdon tubes) possessing predictable spring characteristics. Since pressure 
must accurately translate to motion in a motion-balance system, there must be a predictable 
relationship between pressure and motion in order for the instrument to maintain accuracy. If 
anything happens to affect this pressure/motion relationship such as metal fatigue or temperature 
change, the instrument's calibration will drift. Since there is negligible motion in a force-balance 
system, pressure element spring characteristics are irrelevant to the operation of these devices, and 
their calibrations remain more stable over time. 

Both force- and motion-balance pneumatic instruments are usually equipped with an amplifying 



relay between the nozzle backpressure chamber and the feedback bellows. The purpose of an 
amplifying relay in a self-balancing pneumatic system is the same as the purpose of providing 
an operational amplifier with an extremely high open-loop voltage gain: the more internal gain 
the system has, the closer to ideal the "balancing" effect will be. In other words, our "simplifying 
assumption" of zero baffle/nozzle gap change will be closer to the truth in a system where the nozzle 
pressure gets amplified before going to the feedback bellows: 

Air supply 

>~ ' 

Supply Input 1 

r out 


Output 1 L 



Thus, adding a relay to a self-balancing pneumatic system is analogous to increasing the open- 
loop voltage gain of an opamp (Aol) by several- fold: it makes the overall gain closer to ideal. The 
overall gain of the system, though, is dictated by the ratio of bellows leverage on the force beam, 
just like the overall gain of a negative-feedback opamp circuit is dictated by the feedback network 
and not by the opamp's internal (open-loop) voltage gain. 


9.5 Analysis of a practical pneumatic instrument 

Perhaps one of the most popular pneumatic industrial instruments ever manufactured is the Foxboro 
model 13 differential pressure transmitter. A photograph of one with the cover removed is shown 

The following is a functional illustration of this instrument: 



High pressure 
input — 

, Air 

_ Output 

Zero screw 

Diaphragm seal 

Low pressure 
— input 

Part of the reason for this instrument's popularity is the extreme utility of differential pressure 
transmitters in general. A "DP cell" may be used to measure pressure, vacuum, pressure differential, 
liquid level, liquid or gas flow, and even liquid density. A reason for this particular differential 



transmitter's popularity is excellent design: the Foxboro model 13 transmitter is rugged, easy to 
calibrate, and quite accurate. 

Like so many pneumatic instruments, the model 13 transmitter uses the force-balance (more 
precisely, the motion-balance) principle whereby any shift in position is sensed by a detector (the 
baffle/nozzle assembly) and immediately corrected through negative feedback to restore equilibrium. 
As a result, the output air pressure signal becomes an analogue of the differential process fluid 
pressure sensed by the diaphragm capsule. In the following photograph you can see my index finger 
pointing to the baffle/nozzle mechanism at the top of the transmitter: 

Let's analyze the behavior of this transmitter step- by-step as it senses an increasing pressure 
on the "High pressure" input port. As the pressure here increases, the large diaphragm capsule is 
forced to the right. The same effect would occur if the pressure on the "Low pressure" input port 
were to decrease. This is a differential pressure transmitter, so what it responds to is changes in 
pressure difference between the two input ports. 

This resultant motion of the capsule tugs on the thin flexure connecting it to the force bar. The 
force bar pivots at the fulcrum (where the small diaphragm seal is located) in a counter-clockwise 
rotation, tugging the flexure at the top of the force bar. This motion causes the range bar to also 
pivot at its fulcrum (the sharp-edged "range wheel"), moving the baffle closer to the nozzle. 

As the baffle approaches the nozzle, air flow through the nozzle becomes more restricted, 
accumulating backpressure in the nozzle. This backpressure increase is greatly amplified in the 
relay, which sends an increasing air pressure signal both to the output line and to the bellows at the 
bottom of the range bar. Increasing pneumatic pressure in the bellows causes it to push harder on 
the bottom of the range bar, counterbalancing the initial motion and returning the range bar (and 
force bar) to their near-original positions. 

Calibration of this instrument is accomplished through two adjustments: the zero screw and 
the range wheel. The zero screw simply adds tension to the bottom of the range bar, pulling it in 
such a direction as to collapse the bellows as the zero screw is turned clockwise. This action pushes 



the baffle closer to the nozzle and tends to increase air pressure to the bellows as the system seeks 
equilibrium. If a technician turns the range wheel, the lever ratio of the range bar changes, affecting 
the ratio of force bar force to bellows force. The following photograph shows the range bar and 
range wheel of the instrument: 

As in all instruments, the zero adjustment works by adding or subtracting a quantity, while the 
span adjustment works by multiplying or dividing a quantity. In the Foxboro model 13 pneumatic 
transmitter, the quantity in question is force. The zero screw adds or subtracts force to the 
mechanical system by tensioning a spring, while the range wheel multiplies or divides force in the 


system by changing the mechanical advantage (force ratio) of a lever. 


9.6 Proper care and feeding of pneumatic instruments 

Perhaps the most important rule to obey when using pneumatic instruments is to maintain clean 
and dry instrument air. Compressed air containing dirt, rust, oil, water, or other contaminants will 
cause operational problems for pneumatic instruments. First and foremost is the concern that tiny 
orifices and nozzles inside the pneumatic mechanisms will clog over time. Clogged orifices tend to 
result in decreased output pressure, while clogged nozzles tend to result in increased output pressure. 
In either case, the "first aid" repair is to pass a welding torch tip cleaner through the plugged hole 
to break loose the residue or debris plugging it. 

Moisture in compressed air tends to corrode metal parts inside pneumatic mechanisms. This 
corrosion may break loose to form debris that plugs orifices and nozzles, or it may simply eat 
through thin diaphragms and bellows until air leaks develop. Grossly excessive moisture will cause 
erratic operation as "plugs" of liquid travel through thin tubes, orifices, and nozzles designed only 
for air passage. 

A common mistake made when installing pneumatic instruments is to connect them to a general- 
service ( "utility" ) compressed air supply instead of a dedicated instrument-service compressed air 
system. Utility air systems are designed to supply air tools and large air-powered actuators with 
pneumatic power. These high-flow compressed air systems are often seeded with antifreeze and/or 
lubricating chemicals to prolong the operating life of the piping and air-consuming devices, but 
the same liquids will wreak havoc on sensitive instrumentation. Instrument air supplies should be 
sourced by their own dedicated air compressor(s), complete with automatic air-dryer equipment, 
and distributed through stainless steel, copper, or plastic tubing (never black iron or galvanized iron 

The worst example of moisture in an instrument air system I have ever witnessed is an event 
that happened at an oil refinery where I worked as an instrument technician. Someone on the 
operations staff decided they would use 100 PSI instrument air to purge a process pipe filled with 
acid. Unfortunately, the acid pressure in the process pipe exceeded 100 PSI, and as a result acid 
flushed backward into the instrument air system. Within days most of the pneumatic instruments 
in that section of the refinery failed due to accelerated corrosion of brass and aluminum components 
inside the instruments. The total failure of multiple instruments over such a short time could have 
easily resulted in a disaster, but fortunately the crisis was minimal. Once the first couple of faulty 
instruments were disassembled after removal, the cause of failure became evident and the technicians 
took action to purge the lines of acid before too many more instruments suffered the same fate. 

Pneumatic instruments must be fed compressed air of the proper pressure as well. Just like 
electronic circuits which require power supply voltages within specified limits, pneumatic instruments 
do not operate well if their air supply pressure is too low or too high. If the supply pressure is too 
low, the instrument cannot generate a full-scale output signal. If the supply pressure is too high, 
internal failure may result from ruptured diaphragms, seals, or bellows. Many pneumatic instruments 
are equipped with their own local pressure regulators directly attached to ensure each instrument 
receives the correct pressure despite pressure fluctuations in the supply line. 

Another "killer" of pneumatic instruments is mechanical vibration. These are precision 
mechanical devices, so they do not generally respond well to repeated shaking. At the very least, 
calibration adjustments may loosen and shift, causing the instrument's accuracy to suffer. At worst, 
actual failure may result from component breakage 2 . 

2 Having said this, pneumatic instruments can be remarkably rugged devices. I once worked on a field-mounted 


9.7 Advantages and disadvantages of pneumatic instruments 

The disadvantages of pneumatic instruments are painfully evident to anyone familiar with both 
pneumatic and electronic instruments. Sensitivity to vibration, changes in temperature, mounting 
position, and the like affect calibration accuracy to a far greater degree for pneumatic instruments 
than electronic instruments. Compressed air is an expensive utility - much more expensive per 
equivalent watt-hour than electricity - making the operational cost of pneumatic instruments far 
greater than electronic. The installed cost of pneumatic instruments can be quite high as well, given 
the need for special (stainless steel, copper, or tough plastic) tubes to carry supply air and pneumatic 
signals to distant locations. The volume of air tubes used to convey pneumatic signals over distances 
acts as a low-pass filter, naturally damping the instrument's response and thereby reducing its ability 
to respond quickly to changing process conditions. Pneumatic instruments cannot be made "smart" 
like electronic instruments, either. With all these disadvantages, one might wonder why pneumatic 
instruments are still used at all in modern industry. 

Part of the answer is legacy. For an industrial facility built decades ago, it makes little sense 
to replace instruments that still work just fine. The cost of labor to remove old tubing, install 
new conduit and wires, and configure new (expensive) electronic instruments often is not worth the 

However, pneumatic instruments actually enjoy some definite technical advantages which secure 
their continued use in certain applications even in the 21 s * century. One decided advantage is the 
intrinsic safety of pneumatic field instruments. Instruments that do not run on electricity cannot 
generate electrical sparks. This is of utmost importance in "classified" industrial environments where 
explosive gases, liquids, dusts, and powders exist. Pneumatic instruments are also self-purging. 
Their continual bleeding of compressed air from vent ports in pneumatic relays and nozzles acts as a 
natural clean-air purge for the inside of the instrument, preventing the intrusion of dust and vapor 
from the outside with a slight positive pressure inside the instrument case. It is not uncommon to 
find a field-mounted pneumatic instrument encrusted with corrosion and filth on the outside, but 
factory-clean on the inside due to this continual purge of clean air. Pneumatic instruments mounted 
inside larger enclosures with other devices tend to protect them all by providing a positive-pressure 
air purge for the entire enclosure. 

Some pneumatic instruments can also function in high-temperature and high-radiation 
environments that would damage electronic instruments. Although it is often possible to "harden" 
electronic field instruments to such harsh conditions, pneumatic instruments are practically immune 
by nature. 

An interesting feature of pneumatic instruments is that they may operate on compressed gases 
other than air. This is an advantage in remote natural gas installations, where the natural gas 
itself is sometimes used as a source of pneumatic "power" for instruments. So long as there is 
compressed natural gas in the pipeline to measure and to control, the instruments will operate. No 
air compressor or electrical power source is needed in these installations. What is needed, however, 
is good filtering equipment to prevent contaminants in the natural gas (dirt, debris, liquids) from 

pneumatic controller attached to the same support as a badly cavitating control valve. The vibrations of the control 
valve transferred to the controller through the support, causing the baffle to hammer repeatedly against the nozzle 
until the nozzle's tip had been worn down to a flattened shape. Remarkably, the only indication of this problem 
was the fact the controller was having some difficulty maintaining setpoint. Other than that, it seemed to operate 
adequately! I doubt any electronic device would have fared as well, unless completely "potted" in epoxy. 


causing problems within the sensitive instrument mechanisms. 


Patrick, Dale R. and Patrick, Steven R., Pneumatic Instrumentation, Delmar Publishers, Inc., 
Albany, NY, 1993. 

Chapter 10 

Digital electronic instrumentation 



10.1 The HART digital/analog hybrid standard 

A technological advance introduced in the late 1980's was HART, an acronym standing for Highway 
Addressable Remote Transmitter. The purpose of the HART standard was to create a way for 
instruments to digitally communicate with one another over the same two wires used to convey a 
4-20 mA analog instrument signal. In other words, HART is a hybrid communication standard, with 
one variable (channel) of information communicated by the analog value of a 4-20 mA DC signal, and 
another channel for digital communication whereby many other variables could be communicated 
using pulses of current to represent binary bit values of and 1 . 

The HART standard was developed with existing installations in mind. The medium for digital 
communication had to be robust enough to travel over twisted-pair cables of very long length and 
unknown characteristic impedance. This meant that the data communication rate for the digital 
data had to be very slow, even by 1980's standards. 

Digital data is encoded in HART using the Bell 202 modem standard: two audio-frequency 
"tones" (1200 Hz and 2200 Hz) are used to represent the binary states of "1" and "0," respectively, 
transmitted at a rate of 1200 bits per second. This is known as frequency-shift keying, or FSK. The 
physical representation of these two frequencies is an AC current of 1 mA peak-to-peak superimposed 
on the 4-20 mA DC signal. Thus, when a HART-compatible device "talks" digitally on a two-wire 
loop circuit, it produces tone bursts of AC current at 1.2 kHz and 2.2kHz. The receiving HART 
device "listens" for these AC current frequencies and interprets them as binary bits. 

An important consideration in HART current loops is that the total loop resistance (precision 
resistor values plus wire resistance) must fall within a certain range: 250 ohms to 1100 ohms. Most 
4-20 mA loops (containing a single 250 ohm resistor for converting 4-20 mA to 1-5 V) measure in at 
just over 250 ohms total resistance, and work quite well with HART. Even loops containing two 250 
ohm precision resistors meet this requirement. Where technicians often encounter problems is when 
they set up a loop-powered HART transmitter on the test bench with a lab-style power supply and 
no 250 ohm resistor anywhere in the circuit: 



_ Power 

The HART transmitter may be modeled as two parallel current sources: one DC and one AC. The 
DC current source provides the 4-20 mA regulation necessary to represent the process measurement 
as an analog current value. The AC current source turns on and off as necessary to "inject" the 1 mA 
P-P audio-tone HART signal along the two wires. Inside the transmitter is also a HART modem for 
interpreting AC voltage tones as HART data packets. Thus, data transmission takes place through 
the AC current source, and data reception takes place through a voltage-sensitive modem, all inside 
the transmitter, all "talking" along the same two wires that carry the DC 4-20 mA signal. 

For ease of connection in the field, HART devices are designed to be connected in parallel with 



each other. This eliminates the need to break the loop and interrupt the DC current signal every 
time we wish to connect a HART communicator device to communicate with the transmitter. A 
typical HART communicator may be modeled as another AC current source (along with another 
HART voltage-sensitive modem for receiving HART data). Connected in parallel with the HART 
transmitter, the complete circuit looks something like this: 


_ Power 

HART communicator 

The actual hand-held communicator may look like one of these devices: 



With all these sources in the same circuit, it is advisable to use the Superposition Theorem 
for analysis. This involves "turning off" all but one source at a time to see what the effect is for 
each source, then superimposing the results to see what all the sources do when all are working 

We really only need to consider the effects of either AC current source to see what the problem is 
in this circuit with no loop resistance. Consider the situation where the transmitter is sending HART 
data to the communicator. The AC current source inside the transmitter will be active, injecting its 1 
mA P-P audio-tone signal onto the two wires of the circuit. To apply the Superposition Theorem, we 
replace all the other sources with their own equivalent internal resistances (voltage sources become 
"shorts," and current sources become "opens"): 





HART communicator 

The HART communicator is "listening" for those audio tone signals sent by the transmitter's 
AC source, but it "hears" nothing because the DC power supply's equivalent short-circuit prevents 
any significant AC voltage from developing across the two wires. This is what happens when there 
is no loop resistance: no HART device is able to receive data sent by any other HART device. 

The solution to this dilemma is to install a resistance of at least 250 ohms but not greater than 
1100 ohms between the DC power source and all other HART devices, like this: 



HART communicator 

Loop resistance must be at least 250 ohms to allow the 1 mA P-P AC signal to develop enough 
voltage to be reliably detected by the HART modem in the listening device. The upper limit (1100 



ohms) is not a function of HART communication so much as it is a function of the DC voltage 
drop, and the need to maintain a minimum DC terminal voltage at the transmitter for its own 
operation. If there is too much loop resistance, the transmitter will become "starved" of voltage and 
act erratically. In fact, 1100 ohms of loop resistance may even be excessive if the DC power supply 
voltage is too low! 

Loop resistance is also necessary for the HART transmitter to receive data signals transmitted 
by the HART communicator. If we analyze the circuit when the HART communicator's current 
source is active, we get this result: 


250 < R< 1100 




HART communicator 

Without the loop resistance in place, the DC power supply would "short out" the communicator's 
AC current signal just as effectively as it shorted out the transmitter's AC current signal. The 
presence of a loop resistor in the circuit provides a place for an AC voltage to develop in response 
to the AC current injected by the communicator. This AC voltage (across the loop resistor) is seen 
in the diagram as being directly in parallel with the transmitter, where its internal HART modem 
receives the audio tones and processes the data packets. 

Generally manufacturer instructions recommend that HART communicator devices be connected 
in parallel with the HART field instrument, as shown in the above schematic diagrams. However, it 
is also perfectly valid to connect the communicator device directly in parallel with the loop resistor 
like this: 




250 < R< 1100 
-f — AV — T 

_ Power 


HART communicator 

Connected directly in parallel with the loop resistor, the communicator is able to receive 
transmissions from the HART transmitter just fine, as the DC power source acts as a dead short to 
the AC current HART signal and passes it through to the transmitter. 

This is nice to know, as it is often easier to achieve an alligator-clip connection across the leads 
of a resistor than it is to clip in parallel with the loop wires when at a terminal strip or at the 
controller end of the loop circuit. 

HART technology has given a new lease on the venerable 4-20 mA analog instrumentation signal 
standard. It has allowed new features and capabilities to be added on to existing analog signal loops 
without having to upgrade wiring or change all instruments in the loop. Some of the features of 
HART are listed here: 

• Diagnostic data may be transmitted by the field device (self-test results, out-of-limit alarms, 
preventative maintenance alerts, etc.) 

• Field instruments may be re-ranged remotely through the use of HART communicators 

• Technicians may use HART communicators to force field instruments into different "manual" 
modes for diagnostic purposes (e.g. forcing a transmitter to output a fixed current so as 
to check calibration of other loop components, manually stroking a valve equipped with a 
HART-capable positioner) 

• Field instruments may be programmed with identification data (e.g. tag numbers 
corresponding to plant-wide instrument loop documentation) 



10.1.1 HART multidrop mode 

The HART standard also supports a mode of operation that is totally digital, and capable of 
supporting multiple HART instruments on the same pair of wires. This is known as multidrop 

Every HART instrument has an address number, which is typically set to a value of zero (0). A 
network address is a number used to distinguish one device from another on a broadcast network, 
so messages broadcast across the network may be directed to specific destinations. When a HART 
instrument operates in digital/analog hybrid mode, where it must have its own dedicated wire pair 
for communicating the 4-20 mA DC signal between it and an indicator or controller, there is no 
need for a digital address. An address becomes necessary only when multiple devices are connected 
to the same network wiring, and there arises a need to digitally distinguish one device from another 
on the same network. 

This is a functionality the designers of HART intended from the beginning, although it is 
frequently unused in industry. Multiple HART instruments may be connected directly in parallel 
with one another along the same wire pair, and information exchanged between those instruments 
and a host system, if the HART address numbers are set to non-zero values (between 1 and 15): 

HART communicator 
or PC w/ HART modem 



r~vJ-L/T~ , i ' ' O '' O r~> I I r~^ 

Address 4 Address 1 3 Address 1 Address 5 

Setting an instrument's HART address to a non-zero value is all that is necessary to engage 
multidrop mode. The address numbers themselves are irrelevant, as long as they fall within the 
range of 1 to 15 and are unique to that network. 

The major disadvantage of using HART instruments in multidrop mode is its slow speed. 
Due to HART's slow data rate (1200 bits per second), it may take several seconds to access a 
particular instrument's data on a multidropped network. For some applications such as temperature 
measurement, this slow response time may be acceptable. For inherently faster processes such as 
liquid flow control, it would not be nearly fast enough to provide up-to-date information for the 
control system to act upon. 



10.1.2 HART mult i- variable transmitters 

Some "smart" instruments have the ability to report multiple process variables. A good example 
of this is Coriolis-effect flowmeters, which by their very nature simultaneously measure the density, 
flow rate, and temperature of the fluid passing through them. A single pair of wires can only convey 
one 4-20 mA analog signal, but that same pair of wires may convey multiple digital signals encoded 
in the HART protocol. Digital signal transmission is required to realize the full capability of such 
"multi- variable" transmitters. 

If the host system receiving the transmitter's signal(s) is HART-ready, it may digitally poll the 
transmitters for all variables. If, however, the host system does not "talk" using the HART protocol, 
some other means must be found to "decode" the wealth of digital data coming from the multi- 
variable transmitter. One such device is Rosemount's model 333 HART "Tri-Loop" demultiplexer 
shown in the following photograph: 

This device polls the multi-variable transmitter and converts up to three HART variables into 
independent 4-20 mA analog output signals, which any suitable analog indicator or controller device 
may receive. 

It should be noted that the same caveat applicable to multidrop HART systems (i.e. slow 
speed) applies to HART polling of multi-variable transmitters. HART is a relatively slow digital 
bus standard, and as such it should never be considered for applications demanding quick response. 
In applications where speed is not a concern, however, it is a very practical solution for acquiring 
multiple channels of data over a single pair of wires. 


10.2 Fieldbus standards 

The general definition of a fieldbus is any digital network designed to interconnect field-located 
instruments. By this definition, HART multidrop is a type of industrial fieldbus. However, HART is 
too slow to function as a practical fieldbus for many applications, so other fieldbus standards exist. 
Here is a list showing many popular fieldbus standards: 

• FOUNDATION Fieldbus 

• Profibus PA 

• Profibus DP 

• Profibus FMS 

• Modbus 


• CANbus 


• DeviceNet 


The utility of digital "fieldbus" instruments becomes apparent through the host system these 
instruments are connected to (typically a distributed control system, or DCS). Fieldbus-aware host 
systems usually have means to provide instrument information (including diagnostics) in very easy- 
to-navigate formats. For example, the following screenshot shows the field instrument devices 
connected to a small-scale DCS used in an educational lab. Each instrument appears as an icon, 
which may be explored further simply by pointing-and-clicking with the mouse 1 : 

1 The host system in this case is an Emerson DeltaV DCS, and the device manager software is Emerson AMS. 



AMS Suite: Intelligent Device Manager - [Device Connection View] 

fa File Edit View Tools Window Help 

&Mli] ^]iH _oj mi j4]^j *?] 

AMS Device Manager 


Ik Plant Database 

DeltaV Network 1 

f jj Controller - CTLR-01 

PI A^ I/O System - DeltaV 

I/O HART Card - C04 


^ ! CH03 
^jjjk ! CH04 
i% I/O Fieldbus Card - C02 
^ Fieldbus Port - P01 

10.3 Wireless instrumentation 

At the time of this writing, several manufacturers have developed radio-based process transmitters 
capable of establishing "mesh" networks with each other for the exchange and relaying of digital 
information. These transmitters are battery-powered, which means they have no need for field wiring: 
simply connect them to the process! No clear "winner" has emerged as the technical standard for 
wireless data exchange in a process environment, however. Such technology has the potential to 
revolutionize the industry so long as the problems of data security and operational reliability may 
be adequately addressed. 



HART Communications, Technical Information L452 EN; SAMSON AG 

Chapter 11 

Instrument calibration 

11.1 The meaning of calibration 

Every instrument has at least one input and one output. For a pressure sensor, the input would be 
some fluid pressure and the output would (most likely) be an electronic signal. For a loop indicator, 
the input would be a 4-20 mA current signal and the output would be a human-readable display. 
For a variable-speed motor drive, the input would be an electronic signal and the output would be 
electric power to the motor. 

To calibrate an instrument means to check and adjust (if necessary) its response so that the 
output accurately corresponds to its input throughout a specified range. In order to do this, one 
must expose the instrument to an actual input stimulus of precisely known quantity. For a pressure 
gauge, indicator, or transmitter, this would mean subjecting the pressure instrument to known fluid 
pressures and comparing the instrument response against those known pressure quantities. One 
cannot perform a true calibration without comparing an instrument's response to known stimuli. 

To range an instrument means to set the lower and upper range values so that it responds with 
the desired sensitivity to changes in input. For example, a pressure transmitter set to a range of 
to 200 PSI could be re-ranged to respond on a scale of to 150 PSI. 

In analog instruments, re-ranging could (usually) only be accomplished by re-calibration, since 
the same adjustments were used to achieve both purposes. In digital instruments, calibration and 
ranging are typically separate adjustments, so it is important to know the difference. 




11.2 Zero and span adjustments (analog transmitters) 

The purpose of calibration is to ensure the input and output of an instrument correspond to one 
another predictably throughout the entire range of operation. We may express this expectation in 
the form of a graph, showing how the input and output of an instrument should relate: 

URV 100% 










This graph shows how any given percentage of input should correspond to the same percentage 
of output, all the way from 0% to 100%. 

Things become more complicated when the input and output axes are represented by units of 
measurement other than "percent." Take for instance a pressure transmitter, a device designed to 
sense a fluid pressure and output an electronic signal corresponding to that pressure. Here is a graph 
for a pressure transmitter with an input range of to 100 pounds per square inch (PSI) and an 
electronic output signal range of 4 to 20 milliamps (mA) electric current: 



URV 20 mA 

Output 12mA 






LRV 4 mA 


PSI 50 PSI 100 PSI 

LRV Input pressure URV 

Although the graph is still linear, zero pressure does not equate to zero current. This is called 
a live zero, because the 0% point of measurement (0 PSI fluid pressure) corresponds to a non-zero 
("live") electronic signal. PSI pressure may be the LRV (Lower Range Value) of the transmitter's 
input, but the LRV of the transmitter's output is 4 mA, not mA. 

Any linear, mathematical function may be expressed in "slope-intercept" equation form: 

y = mx + b 


y = Vertical position on graph 

x = Horizontal position on graph 

m = Slope of line 

b = Point of intersection between the line and the vertical (y) axis 

This instrument's calibration is no different. If we let x represent the input pressure in units 
of PSI and y represent the output current in units of milliamps, we may write an equation for this 
instrument as follows: 

y = 0.16a; + 4 

On the actual instrument (the pressure transmitter), there are two adjustments which let us 
match the instrument's behavior to the ideal equation. One adjustment is called the zero while 
the other is called the span. These two adjustments correspond exactly to the b and m terms of 
the linear function, respectively: the "zero" adjustment shifts the instrument's function vertically 
on the graph, while the "span" adjustment changes the slope of the function on the graph. By 
adjusting both zero and span, we may set the instrument for any range of measurement within the 
manufacturer's limits. 


It should be noted that for most analog instruments, these two adjustments are interactive. That 
is, adjusting one has an effect on the other. Specifically, changes made to the span adjustment almost 
always alter the instrument's zero point. An instrument with interactive zero and span adjustments 
requires much more effort to accurately calibrate, as one must switch back and forth between the 
lower- and upper-range points repeatedly to adjust for accuracy. 



11.3 LRV and URV settings, digital trim (digital 

The advent of "smart" field instruments containing microprocessors has been a great advance for 
industrial instrumentation. These devices have built-in diagnostic ability, greater accuracy (due to 
digital compensation of sensor nonlinearities), and the ability to communicate digitally with host 
devices for reporting of various parameters. 

A simplified block diagram of a "smart" pressure transmitter looks something like this: 

"Smart" pressure transmitter 

Apply pressure 

[ilM I 

Range adjustments 



Trim adjustments 

Low High 





Trim adjustments 






4-20 mA 


It is important to note all the adjustments within this device, and how this compares to the 
relative simplicity of an all-analog pressure transmitter: 



Analog pressure transmitter 

Calibration adjustments 

Zero Span 

Apply pressure 



[7777T =5 

1 m 





4-20 mA 

Note how the only calibration adjustments available in the analog transmitter are the "zero" and 
"span" settings. Not so with the smart transmitter! Not only can we set lower- and upper-range 
values (LRV and URV), but it is also possible to calibrate the analog-to-digital and digital-to- 
analog converter circuits independently! What this means for the calibration technician is that a 
full calibration procedure on a smart transmitter will potentially require more work and a greater 
number of adjustments than an all-analog transmitter! 

A common mistake made among students and experienced technicians alike is to confuse the 
range settings (LRV and URV) for actual calibration adjustments. Just because you digitally set the 
LRV of a pressure transmitter to 0.00 PSI and the URV to 100.00 PSI does not necessarily mean it 
will register accurately at points within that range! The following example will illustrate this fallacy. 

Suppose we have a smart pressure transmitter ranged for to 100 PSI with an analog output 
range of 4 to 20 mA, but this transmitter's pressure sensor is fatigued from years of use such that an 
actual applied pressure of 100 PSI generates a signal that the analog-to-digital converter interprets 
as only 96 PSI. Assuming everything else in the transmitter is in perfect condition, with perfect 
calibration, the output signal will still be in error: 



"Smart" pressure transmitter 

Range adjustments 

100 PSI 
applied I 


Trim adjustments 



analog signal ^ 

, \ 





100 PSI Trim adjustments 

Low High 







96.00 PSI 

digital value 

19.36 mA 

digital value 


19.36 mA 

As the saying goes, "a chain is only as strong as its weakest link." Here we see how the 
calibration of a sophisticated pressure transmitter may be corrupted despite perfect calibration 
of both analog/digital converter circuits, and perfect range settings in the microprocessor. The 
microprocessor "thinks" the applied pressure is only 96 PSI, and it responds accordingly with a 
19.36 mA output signal. The only way anyone would ever know this transmitter was inaccurate at 
100 PSI is to actually apply a known value of 100 PSI fluid pressure to the sensor and note the 
incorrect response. The lesson here should be clear: digitally setting a smart instrument's LRV and 
URV points does not constitute a legitimate calibration of the instrument. 

For this reason, smart instruments always provide a means to perform what is called a digital trim 
on both the ADC and DAC circuits, to ensure the microprocessor "sees" the correct representation 
of the applied stimulus and to ensure the microprocessor's output signal gets accurately converted 
into a DC current, respectively. 

I have witnessed some technicians use the LRV and URV settings in a manner not unlike the 
zero and span adjustments on an analog transmitter to correct errors such as this. Following this 
methodology, we would have to set the URV of the fatigued transmitter to 96 PSI instead of 100 
PSI, so that an applied pressure of 100 PSI would give us the 20 mA output signal we desire. In 
other words, we would let the microprocessor "think" it was only seeing 96 PSI, then skew the URV 
so that it output the correct signal anyway. Such an approach will work to an extent, but any digital 


queries to the transmitter (e.g. using a digital-over-analog protocol such as HART) will result in 
conflicting information, as the current signal represents full scale (100 PSI) while the digital register 
inside the transmitter shows 96 PSI. The only comprehensive solution to this problem is to "trim" 
the analog-to-digital converter so that the transmitter's microprocessor "knows" the actual pressure 
value applied to the sensor. 

Once digital trims have been performed on both input and output converters, of course, the 
technician is free to re-range the microprocessor as many times as desired without re-calibration. 
This capability is particularly useful when re-ranging is desired for special conditions, such as process 
start-up and shut-down when certain process variables drift into uncommon regions. An instrument 
technician may use a hand-held digital "communicator" device to re-set the LRV and URV range 
values to whatever new values are desired by operations staff without having to re-check calibration 
by applying known physical stimuli to the instrument. So long as the ADC and DAC trims are 
both fine, the overall accuracy of the instrument will still be good with the new range. With analog 
instruments, the only way to switch to a different measurement range was to change the zero and 
span adjustments, which necessitated the re- application of physical stimuli to the device (a full re- 
calibration). Here and here alone we see where calibration is not necessary for a smart instrument. If 
overall measurement accuracy must be verified, however, there is no substitute for an actual physical 
calibration, and this entails both ADC and DAC "trim" procedures for a smart instrument. 


11.4 Calibration procedures 

11.4.1 Linear instruments 

The simplest calibration procedure for a linear instrument is the so-called zero-and-span method. 
The method is as follows: 

1. Apply the lower-range value stimulus to the instrument, wait for it to stabilize 

2. Move the "zero" adjustment until the instrument registers accurately at this point 

3. Apply the upper-range value stimulus to the instrument, wait for it to stabilize 

4. Move the "span" adjustment until the instrument registers accurately at this point 

5. Repeat steps 1 through 4 as necessary to achieve good accuracy at both ends of the range 

An improvement over this crude procedure is to check the instrument's response at several points 
between the lower- and upper-range values. A common example of this is the so-called five-point 
calibration where the instrument is checked at 0% (LRV), 25%, 50%, 75%, and 100% (URV) of 
range. A variation on this theme is to check at the five points of 10%, 25%, 50%, 75%, and 90%, 
while still making zero and span adjustments at 0% and 100%. Regardless of the specific percentage 
points chosen for checking, the goal is to ensure that minimum accuracy is maintained at all points 
along the scale, so that the instrument's response may be trusted when placed into service. 

Yet another improvement over the basic five-point test is to check the instrument's response 
at five calibration points decreasing as well as increasing. Such tests are often referred to as Up- 
down calibrations. The purpose of such a test is to determine if the instrument has any significant 
hysteresis: a lack of responsiveness to a change in direction. 

Some linear instruments provide a means to adjust linearity. This adjustment should be moved 
only if absolutely necessary! Quite often, these linearity adjustments are very sensitive, and prone 
to over-adjustment by zealous fingers. The linearity adjustment of an instrument should be changed 
only if the required accuracy cannot be achieved across the full range of the instrument. Otherwise, 
it is advisable to adjust the zero and span controls to "split" the error between the highest and 
lowest points on the scale, and leave linearity alone. 

11.4.2 Nonlinear instruments 

The calibration of inherently nonlinear instruments is much more challenging than for linear 
instruments. No longer are two adjustments (zero and span) sufficient, because more than two 
points are necessary to define a curve. 

Examples of nonlinear instruments include expanded-scale electrical meters, square root 
characterizers, and position-characterized control valves. 

Every nonlinear instrument will have its own recommended calibration procedure, so I will defer 
you to the manufacturer's literature for your specific instrument. I will, however, offer one piece 
of advice. When calibrating a nonlinear instrument, document all the adjustments you make (e.g. 
how many turns on each calibration screw) just in case you find the need to "re-set" the instrument 
back to its original condition. More than once I have struggled to calibrate a nonlinear instrument 
only to find myself further away from good calibration than where I originally started. In times like 
these, it is good to know you can always reverse your steps and start over! 


11.4.3 Discrete instruments 

The word "discrete" means individual or distinct. In engineering, a "discrete" variable or 
measurement refers to a true-or-false condition. Thus, a discrete sensor is one that is only able 
to indicate whether the measured variable is above or below a specified setpoint. 

Examples of discrete instruments are process switches designed to turn on and off at certain 
values. A pressure switch, for example, used to turn an air compressor on if the air pressure ever 
falls below 85 PSI, is an example of a discrete instrument. 

Discrete instruments need regular calibration just like continuous instruments. Most discrete 
instruments have but one calibration adjustment: the set-point or trip-point. Some process switches 
have two adjustments: the set-point as well as a deadband adjustment. The purpose of a deadband 
adjustment is to provide an adjustable buffer range that must be traversed before the switch changes 
state. To use our 85 PSI low air pressure switch as an example, the set-point would be 85 PSI, but 
if the deadband were 5 PSI it would mean the switch would not change state until the pressure rose 
above 90 PSI (85 PSI + 5 PSI). 

When calibrating a discrete instrument, you must be sure to check the accuracy of the set-point 
in the proper direction of stimulus change. For our air pressure switch example, this would mean 
checking to see that the switch changed states at 85 PSI falling, not 85 PSI rising. If it were not for 
the existence of deadband, it would not matter which way the applied pressure changed during the 
calibration test. However, deadband will always be present in a discrete instrument, whether that 
deadband is adjustable or not. Given a deadband of 5 PSI for this example switch, the difference 
between verifying a change of state at 85 PSI falling versus 85 PSI rising would mean the difference 
between the air compressor turning on if the pressure fell below 85 PSI versus turning on if the 
pressure fell below 80 PSI. 

A procedure to efficiently calibrate a discrete instrument without too many trial-and-error 
attempts is to set the stimulus at the desired value (e.g. 85 PSI for our hypothetical low-pressure 
switch) and then move the set-point adjustment in the opposite direction as the intended direction 
of the stimulus (in this case, increasing the set-point value until the switch changes states). The 
basis for this technique is the realization that most comparison mechanisms cannot tell the difference 
between a rising process variable and a falling setpoint (or visa- versa). Thus, a falling pressure may 
be simulated by a rising set-point adjustment. You should still perform an actual changing-stimulus 
test to ensure the instrument responds properly under realistic circumstances, but this "trick" will 
help you achieve good calibration in less time. 


11.5 Typical calibration errors 

Recall that the slope- intercept form of a linear equation describes the response of a linear instrument: 

y = mx + b 


y = Output 

m = Span adjustment 

X = Input 

b = Zero adjustment 

A zero shift calibration error shifts the function vertically on the graph. This error affects all 
calibration points equally, creating the same percentage of error across the entire range: 

20 mA 

12 mA 


4 mA 








/ 1 



f / 

















Y = 

- m 






Input pressure 

100 PSI 

A span shift calibration error shifts the slope of the function. This error's effect is unequal at 
different points throughout the range: 



20 mA 

12 mA 


4 mA 




















ie efiecofa soan shift 

y = mx + b 



Input pressure 

100 PSI 

A linearity calibration error causes the function to deviate from a straight line. This type of 
error does not directly relate to a shift in either zero (b) or span (m) because the slope-intercept 
equation only describes straight lines. If an instrument does not provide a linearity adjustment, the 
best you can do for this type of error is "split the error" between high and low extremes, so that the 
maximum absolute error at any point in the range is minimized: 

20 mA 

12 mA 


4 mA 



V 1 




























The effect of a lint 






Input pressure 

100 PSI 



A hysteresis calibration error occurs when the instrument responds differently to an increasing 
input compared to a decreasing input. The only way to detect this type of error is to do an up-down 
calibration test, checking for instrument response at the same calibration points going down as going 

20 mA 

12 mA 


4 mA - 
























he arro 










of moti 



Input pressure 

100 PSI 

Hysteresis errors are almost always caused by mechanical friction on some moving element 
(and/or a loose coupling between mechanical elements) such as bourdon tubes, bellows, diaphragms, 
pivots, levers, or gear sets. Flexible metal strips called flexures - which are designed to serve as 
frictionless pivot points in mechanical instruments - may also cause hysteresis errors if cracked or 

In practice, most calibration errors are some combination of zero, span, linearity, and hysteresis 



11.5.1 As-found and as-left documentation 

An important principle in calibration practice is to document every instrument's calibration as it 
was found and as it was left after adjustments were made. The purpose for documenting both 
conditions is so that data is available to calculate instrument drift over time. If only one of these 
conditions is documented during each calibration event, it will be difficult to determine how well an 
instrument is holding its calibration over long periods of time. Excessive drift is often an indicator 
of impending failure, which is vital for any program of predictive maintenance or quality control. 

Typically, the format for documenting both As-Found and As-Left data is a simple table showing 
the points of calibration, the ideal instrument responses, the actual instrument responses, and the 
calculated error at each point. The following table is an example for a pressure transmitter with a 
range of to 200 PSI over a five-point scale: 

of range 


Output current 

Output current 

(percent of span) 



4.00 mA 


50 PSI 

8.00 mA 


100 PSI 

12.00 mA 


150 PSI 

16.00 mA 


200 PSI 

20.00 mA 

11.5.2 Up-tests and Down-tests 

It is not uncommon for calibration tables to show multiple calibration points going up as well as 
going down, for the purpose of documenting hysteresis and deadband errors. Note the following 
example, showing a transmitter with a maximum hysteresis of 0.313 % (the offending data points 
are shown in bold-faced type): 

of range 


Output current 

Output current 

(percent of span) 



4.00 mA 

3.99 mA 

-0.0625 % 

25% T 

50 PSI 

8.00 mA 

7.98 mA 

-0.125 % 

50% t 

100 PSI 

12.00 mA 

11.99 mA 

-0.0625 % 

75% T 

150 PSI 

16.00 mA 

15.99 mA 

-0.0625 % 

100% t 

200 PSI 

20.00 mA 

20.00 mA 


75% I 

150 PSI 

16.00 mA 

16.01 mA 

+0.0625 % 

50% | 

100 PSI 

12.00 mA 

12.02 mA 

+0.125 % 

25% | 

50 PSI 

8.00 mA 

8.03 mA 

+0.188 % 



4.00 mA 

4.01 mA 

+0.0625 % 

In the course of performing such a directional calibration test, it is important not to overshoot 
any of the test points. If you do happen to overshoot a test point in setting up one of the input 
conditions for the instrument, simply "back up" the test stimulus and re- approach the test point 
from the same direction as before. Unless each test point's value is approached from the proper 
direction, the data cannot be used to determine hysteresis/deadband error. 


11.6 NIST traceability 

As defined previously, calibration means the comparison and adjustment (if necessary) of an 
instrument's response to a stimulus of precisely known quantity, to ensure operational accuracy. 
In order to perform a calibration, one must be reasonably sure that the physical quantity used to 
stimulate the instrument is accurate in itself. For example, if I try calibrating a pressure gauge to 
read accurately at an applied pressure of 200 PSI, I must be reasonably sure that the pressure I am 
using to stimulate the gauge is actually 200 PSI. If it is not 200 PSI, then all I am doing is adjusting 
the pressure gauge to register 200 PSI when in fact it is sensing something different. 

Ultimately, this is a philosophical question of epistemology: how do we know what is true? 
There are no easy answers here, but teams of scientists and engineers known as metrologists devote 
their professional lives to the study of calibration standards to ensure we have access to the best 
approximation of "truth" for our calibration purposes. Metrology is the science of measurement, 
and the central repository of expertise on this science within the United States of America is the 
National Institute of Standards and Technology, or the NIST (formerly known as the National Bureau 
of Standards, or NBS). 

Experts at the NIST work to ensure we have means of tracing measurement accuracy back to 
intrinsic standards, which are quantities inherently fixed (as far as anyone knows). The vibrational 
frequency of an isolated cesium atom when stimulated by radio energy, for example, is an intrinsic 
standard used for the measurement of time (forming the basis of the so-called atomic clock). So far 
as anyone knows, this frequency is fixed in nature and cannot vary. Intrinsic standards therefore 
serve as absolute references which we may calibrate certain instruments against. 

The machinery necessary to replicate intrinsic standards for practical use are quite expensive and 
usually delicate. This means the average metrologist (let alone the average industrial instrument 
technician) simply will never have access to one. In order for these intrinsic standards to be useful 
within the industrial world, we use them to calibrate other instruments, which are used to calibrate 
other instruments, and so on until we arrive at the instrument we intend to calibrate for field service in 
a process. So long as this "chain" of instruments is calibrated against each other regularly enough to 
ensure good accuracy at the end-point, we may calibrate our field instruments with confidence. The 
documented confidence is known as NIST traceability: that the accuracy of the field instrument we 
calibrate is ultimately ensured by a trail of documentation leading to intrinsic standards maintained 
by the NIST. 

11.7 Instrument turndown 

An important performance parameter for transmitter instruments is something often referred to as 
turndown or rangedown. "Turndown" is defined as the ratio of maximum allowable span to the 
minimum allowable span for a particular instrument. 

Suppose a pressure transmitter has a maximum calibration range of to 300 pounds per square 
inch (PSI), and a turndown of 20:1. This means that a technician may adjust the span anywhere 
between 300 PSI and 15 PSI. This is important to know in order to select the proper transmitter 
for any given measurement application. The odds of you finding a transmitter with just the perfect 
factory-calibrated range for your measurement application may be quite small, meaning you will 
have to adjust its range to fit your needs. The turndown ratio tells you how far you will be able to 
practically adjust your instrument's range. 


11.8 Practical calibration standards 

Within the context of a calibration shop environment, where accurate calibrations are important 
yet intrinsic standards are not readily accessible, we must do what we can to maintain a workable 
degree of accuracy in the calibration equipment used to calibrate field instruments. 

It is important that the degree of uncertainty in the accuracy of a test instrument is significantly 
less than the degree of uncertainty we hope to achieve in the instruments we calibrate. Otherwise, 
calibration becomes an exercise in futility. This ratio of uncertainties is called the Test Uncertainty 
Ratio, or TUR. A good rule-of-thumb is to maintain a TUR of at least 4:1 (ideally 10:1 or better), 
the test equipment being many times more accurate (less uncertain) than the field instruments we 
calibrate with them. 

I have personally witnessed the confusion and wasted time that results from trying to calibrate a 
field instrument to a tighter tolerance than what the calibrating equipment is capable of. In one case, 
an instrument technician attempted to calibrate a pneumatic pressure transmitter to a tolerance of 
+/- 0.5% of span using a test gauge that was only good for +/- 1% of the same span. This poor 
technician kept going back and forth, adjusting zero and span over and over again, trying to stay 
within the stated specification of 0.5%. After giving up, he tested the test gauges by comparing 
three of them, one against the other. When it was realized no two test gauges would agree with 
each other to within the tolerance he was trying to achieve in calibrating the transmitter, it became 
clear what the problem was. 

The lesson to be learned here is to always ensure the equipment used to calibrate industrial 
instruments is reliably accurate (enough). No piece of test equipment will ever be perfectly accurate, 
but perfection is not what we need. Our goal is to be accurate enough that the final calibration will 
be reliable within specified boundaries. 

The next few subsections describe various standards used in instrument shops to calibrate 
industrial instruments. 



11.8.1 Electrical standards 

Electrical calibration equipment - used to calibrate instruments measuring voltage, current, and 
resistance - must be periodically calibrated against higher-tier standards maintained by outside 
laboratories. In years past, instrument shops would often maintain their own standard cell batteries 
(often called Weston cells) as a primary voltage reference. These special-purpose batteries produced 
1.0183 volts DC at room temperature with low uncertainty and drift, but were sensitive to vibration 
and non-trivial to actually use. Now, electronic voltage references have all but displaced standard 
cells in calibration shops and laboratories, but these references must be checked and adjusted for 
drift in order to maintain their NIST traceability. 

One enormous benefit of electronic calibration references is that they are able to generate 
accurate currents and resistances in addition to voltage (and not just voltage at one fixed value, 
either!). Modern electronic references are digitally-controlled as well, which lends themselves well 
to automated testing in assembly-line environments, and/or programmed multi-point calibrations 
with automatic documentation of as-found and as-left calibration data. 

If a shop cannot afford one of these versatile references for benchtop calibration use, an acceptable 
alternative in some cases is to purchase a high-accuracy multimeter and equip the calibration bench 
with adjustable voltage, current, and resistance sources. These sources will be simultaneously 
connected to the high-accuracy multimeter and the instrument under test, and adjusted until the 
high-accuracy meter registers the desired value. The measurement shown by the instrument under 
test is then compared against the reference meter and adjusted until matching (to within the required 
tolerance) . The following illustration shows how a high-accuracy voltmeter could be used to calibrate 
a handheld voltmeter in this fashion: 

Handheld multimeter 


n ■*■ 

High-accuracy (benchtop) multimeter 


V A n Con 


Variable voltage source 



iA ~ Fine 



It should be noted that the variable voltage source shown in this test arrangement need not be 
sophisticated. It simply needs to be variable (to allow precise adjustment until the high-accuracy 
voltmeter registers the desired voltage value) and stable (so that the adjustment does not drift 
appreciably over time). 


11.8.2 Temperature standards 

The most common technologies for industrial temperature measurement are electronic in nature: 
RTDs and thermocouples. As such, the standards used to calibrate such devices are the same 
standards used to calibrate electrical instruments such as digital multimeters (DMMs). 

However, there are some temperature-measuring instruments that are not electrical in nature. 
This category includes bimetallic thermometers, filled-bulb temperature systems, and optical 
pyrometers. In order to calibrate these types of instruments, we must accurately create the 
calibration temperatures in the instrument shop. 

A time-honored standard for low-temperature industrial calibrations is water, specifically the 
freezing and boiling points of water. Pure water at sea level (full atmospheric pressure) freezes at 
32 degrees Fahrenheit (0 degrees Celsius) and boils at 212 degrees Fahrenheit (100 degrees Celsius). 
In fact, the Celsius temperature scale is defined by these two points of phase change for water at 
sea level 1 . 

To use water as a temperature calibration standard, simply prepare a vessel for one of two 
conditions: thermal equilibrium at freezing or thermal equilibrium at boiling. "Thermal equilibrium" 
in this context simply means equal temperature throughout the mixed-phase sample. In the case of 
freezing, this means a well-mixed sample of solid ice and liquid water. In the case of boiling, this 
means a pot of water at a steady boil (vaporous steam and liquid water in direct contact). What 
you are trying to achieve here is ample contact between the two phases (either solid and liquid; or 
liquid and vapor) to eliminate hot or cold spots. 

One major disadvantage of using phase changes to produce accurate temperatures in the shop is 
the limited availability of temperatures. With water at sea level, the only calibration standards you 
can create is degrees Celsius and 100 degrees Celsius. If you need to create some other temperature 
for calibration purposes, you either need to find a suitable material with a phase change happening 
at that temperature (good luck!) or you need to find a finely adjustable temperature source and use 
an accurate thermometer to compare your instrument under test against. This scenario is analogous 
to the use of a high-accuracy voltmeter and an adjustable voltage source to calibrate a voltage 

Laboratory-grade thermometers are relatively easy to secure. Variable temperature sources 
suitable for calibration use include oil bath and sand bath calibrators. These devices are exactly 
what they sound like: small pots filled with either oil or sand, containing an electric heating element 
and a temperature control system using a laboratory-grade (NIST-traceable) thermal sensor. In the 
case of sand baths, a small amount of compressed air is introduced at the bottom of the vessel to 
"fluidize" the sand so that the grains move around much like the molecules of a liquid, helping the 
system reach thermal equilibrium. To use a bath-type calibrator, place the temperature instrument 
to be calibrated so that the sensing element dips into the bath, then wait for the bath to reach the 
desired temperature. 

An oil bath temperature calibrator is shown in the following photograph, with sockets to accept 
seven temperature probes into the heated oil reservoir: 

1 The Celsius scale used to be called the Centigrade scale, which literally means "100 steps." I personally prefer 
"Centigrade" to "Celsius" because it actually describes something about the unit of measurement. In the same vein, 
I also prefer the older label "Cycles Per Second" (cps) to "Hertz" as the unit of measurement for frequency. You may 
have noticed by now that the instrumentation world does not yield to my opinions, much to my chagrin. 



Dry-block temperature calibrators also exist for creating accurate calibration temperatures in the 
instrument shop environment. Instead of a fluid (or fluidized powder) bath as the thermal medium, 
these devices use metal blocks with blind (dead-end) holes drilled for the insertion of temperature- 
sensing instruments. 

An inexpensive dry-block temperature calibrator intended for bench-top service is shown in this 



Optical temperature instruments require a different sort of calibration tool: one that emits 
radiation equivalent to that of the process object at certain specified temperatures. This type of 
calibration tool is called a blackbody calibrator, having a target area where the optical instrument 
may be aimed. Like oil and sand bath calibrators, a blackbody calibrator relies on an internal 
temperature sensing element as a reference, to control the optical emissions of the blackbody target 
at any specified temperature within a practical range. 



11.8.3 Pressure standards 

In order to accurately calibrate a pressure instrument in a shop environment, we must create fluid 
pressures of known magnitude against which we compare the instrument being calibrated. As with 
other types of physical calibrations, our choices of instruments falls into two broad categories: devices 
that inherently produce known pressures versus devices that accurately measure pressures created 
by some (other) adjustable source. 

A deadweight tester (sometimes referred to as a dead-test calibrator) is an example in the former 
category. These devices create accurately known pressures by means of precise masses and pistons 
of precise area: 

Deadweight tester 

Gauge to be 




"Primary piston 

A. 'V. 

Oil or water Secondary piston 

After connecting the gauge (or other pressure instrument) to be calibrated, the technician adjusts 
the secondary piston to cause the primary piston to lift off its resting position and be suspended 
by oil pressure alone. So long as the mass placed on the primary piston is precisely known, Earth's 
gravitational field is constant, and the piston is perfectly vertical, the fluid pressure applied to the 
instrument under test must be equal to the value described by the following equation: 




P = Fluid pressure 

F = Force exerted by the action of gravity on the mass (F we i g ht 

A = Area of piston 


The primary piston area, of course, is precisely set at the time of the deadweight tester's 
manufacture and does not change appreciably throughout the life of the device. 

A very simple deadweight tester unit appears in the next photograph, mounted to a yellow 
wooden base: 



When sufficient pressure has been accumulated inside the tester to overcome the weight on the 
piston, the piston rises off its rest and "floats" on the pressurized oil, as shown in this close-up 



A common operating practice for any deadweight tester is to gently spin the mass during testing 
so that the primary piston continually rotates within its cylinder. Any motion will prevent static 
friction from taking hold, helping to ensure the only force on the primary piston is the force of the 
fluid within the deadweight tester. 

Most modern deadweight testers include extra features such as hand pumps and bleed valves in 
addition to secondary pistons, to facilitate both rapid and precise operation. The next photograph 
shows a newer deadweight tester, with these extra features: 

There is also such a thing as a pneumatic deadweight tester. In these devices, a constant flow of 
gas such as compressed air or bottled nitrogen vents through a bleed port operated by the primary 
piston. The piston moves as necessary to maintain just enough gas pressure inside the unit to 
suspend the mass(es) against gravity. This gas pressure passes on to the instrument under test, just 
as liquid pressure in a hydraulic deadweight tester passes to the test instrument for comparison: 



Pneumatic deadweight tester 


Gauge to be 

Primary piston 
~* — Bleed 

From gas 


In fact, the construction and operation of a pneumatic deadweight tester is quite similar to a 
self-balancing (force-balance) pneumatic instrument mechanism with a baffle/nozzle assembly. A 
moving element opens or closes a variable restriction downstream of a fixed restriction to generate 
a varying pressure. In this case, that pressure directly operates the bleed vent to self-regulate gas 
pressure at whatever value is necessary to suspend the mass against gravity. 

Deadweight testers (both hydraulic and pneumatic) lend themselves well to relatively high 
pressures, owing to the practical limitations of mass and piston area. You could use a deadweight 
tester to calibrate a 100 PSI pressure gauge used for measuring water mains pressure, for example, 
but you could not use a deadweight tester to calibrate a to 1 " W.C. (zero to one inch water column) 
pressure gauge used to measure draft pressure in a furnace flue. 

For low-pressure calibrations, the simple manometer is a much more practical standard. 
Manometers, of course, do not generate pressure on their own. In order to use a manometer to 
calibrate a pressure instrument, you must connect both devices to a source of variable fluid pressure, 
typically instrument air through a precision pressure regulator: 



Gauge to be 






From air 

The difference in liquid column heights (h) within the manometer shows the pressure applied to 
the gauge. As with the deadweight tester, the accuracy of this pressure measurement is bound by 
just a few physical constants, none of which are liable to spurious change. So long as the manometer's 
liquid density is precisely known, Earth's gravitational field is constant, and the manometer tubes 
are perfectly vertical, the fluid pressure indicated by the manometer must be equal to the value 
described by the following equation (two different forms given): 

P = P gh 


P = jh 


P = Fluid pressure 

p = Mass density of fluid 

7 = Weight density of fluid 

g = Acceleration of gravity 

h = Height difference between manometer liquid columns 

Of course, with pressure-measuring test instruments of suitable accuracy (preferably NIST- 
traceable), the same sort of calibration jig may be used for virtually any desired range of pressures: 




Gauge to be 


From air 

When the electronic test gauge is designed for very low pressures (inches of water column) , they 
are sometimes referred to as electronic manometers. 

Instrument calibrations performed in the field (i.e. in locations near or at the intended point 
of use rather than in a professionally-equipped shop) are almost always done this way: a pressure- 
generating source is connected to both the instrument under test and a trusted calibration gauge 
("test gauge"), and the two indications are compared at several points along the calibrated range. 
Test equipment suitable for field pressure calibrations include slack-tube manometers made from 
flexible plastic tubing hung from any available anchor point near eye level, and test gauges typically 
of the helical bourdon tube variety. Portable electronic test gauges are also available for field use, 
many with built-in hand pumps for generating precise air pressures. 

A noteworthy example of a pneumatic pressure calibrator for field use was a device manufactured 
by the Wallace & Tiernan corporation, affectionately called a Wally box by at least one generation 
of instrument technicians. A "Wally box" consisted of a large dial pressure gauge (several inches 
in diameter) with a multi-turn needle and a very fine scale, connected to a network of valves and 
regulators which were used to set different air pressures from any common compressed air source. 
The entire mechanism was housed in an impact-resistance case for ruggedness. One of the many 
nice features of this calibration instrument was a selector valve allowing the technician to switch 
between two different pressures output by independent pressure regulators. Once the two pressure 
regulator values were set to the instrument's lower- and upper-range values (LRV and URV), it was 
possible to switch back and forth between those two pressures at will, making the task of adjusting 
an analog instrument with interactive zero and span adjustments much easier than it would have 
been to precisely adjust a single pressure regulator again and again. 


11.8.4 Flow standards 

Most forms of continuous flow measurement are inferential; that is, we measure flow indirectly by 
measuring some other variable (such as pressure, voltage, or frequency) directly. With this in mind, 
we may usually achieve reasonable calibration accuracy simply by calibrating the primary sensor 
and replacing the flow element (if inspection proves necessary) . In the case of an orifice plate used to 
measure fluid flow rate, this would mean calibrating the differential pressure transmitter to measure 
pressure accurately and replacing the orifice plate if it shows signs of wear. 

In some cases, though, direct validation of flow measurement accuracy is needed. Most techniques 
of flow rate validation take the form of measuring accumulated fluid volume over time. This may 
prove to be complicated, especially if the fluids in question are hazardous in any way, and/or the 
flow rates are large, and/or the fluid is a gas or vapor. 

For simple validation of liquid flow rates, the flow may be diverted from its normal path in 
the process and into a container where either accumulated volume or accumulated weight may be 
measured over time. If the rate of flow into this container is constant, the accumulated volume (or 
weight) should increase linearly over time. The actual flow rate may then be calculated by dividing 
the change in volume (AV) by the time interval over which the change in volume was measured 
(At). The resulting quotient is the average flow rate between those two points in time, which is an 
approximation of instantaneous flow rate: 


= Average flow 


AV dV 

ss = Instantaneous flow 

At dt 

If a suitable vessel exists in the process with level-measuring capability (e.g. a liquid storage 
vessel equipped with a level transmitter), you may apply the same mathematical technique: use 
that vessel as an accumulator for the flow in question, tracking the accumulated (or lost) volume 
over time and then calculating -^ . The accuracy of this technique rests on some additional factors, 

• The accuracy of the level transmitter (as a volume measuring instrument!) 

• The ability to ensure only one flow path in or out of that vessel 

The first condition listed here places significant limitations on the flow calibration accuracy one 
can achieve with this method. In essence, you are using the level instrument as the "test gauge" for 
the flow instrument, so it needs to be high- accuracy in order to achieve even reasonable accuracy 
for the flowmeter being calibrated. 

A more sophisticated approach for direct flow validation is the use of a device called a flow 
prover. A "flow prover" is a precision piston-and-cylinder mechanism used to precisely measure a 
quantity of liquid over time. Process flow is diverted through the prover, moving the piston over 
time. Sensors on the prover mechanism detect when the piston has reached certain positions, and 
time measurements taken at those different positions enable the calculation of average flow (-^f )■ 


11.8.5 Analytical standards 

An analyzer measures intrinsic properties of a substance sample such as its density, chemical content, 
or purity. Whereas the other types of instruments discussed in this chapter measure quantities 
incidental to the composition of a substance (pressure, level, temperature, and flow rate), an analyzer 
measures something related to the nature of substance being processed. 

As previously defined, to calibrate an instrument means to check and adjust (if necessary) its 
response so that the output accurately corresponds to its input throughout a specified range. In 
order to do this, one must expose the instrument to an actual input stimulus of precisely known 
quantity. This is no different for an analytical instrument. In order to calibrate an analyzer, we 
must exposed it to known quantities of substances with the desired range of properties (density, 
chemical composition, etc.). 

A classic example of this is the calibration of a pH analyzer. pH is the measurement of hydrogen 
ion activity in an aqueous solution. The standard range of measurement is pH to 14 pH, the 
number representing a negative power of 10 approximately describing the hydrogen ion molarity of 
the solution (how many moles of active hydrogen ions per liter of solution) 2 . 

The pH of a solution is typically measured with a pair of special electrodes immersed in the 
solution, which generate a voltage proportional to the pH of the solution. In order to calibrate 
a pH instrument, you must have a sample of liquid solution with a known pH value. For pH 
instrumentation, such calibration solutions are called buffers, because they are specially formulated 
to maintain stable pH values even in the face of (slight levels of) contamination. 

pH buffers may be purchased in liquid form or in powder form. Liquid buffer solutions may be 
used directly out of the bottle, while powdered buffers must be dissolved in appropriate quantities 
of de-ionized water to generate a solution ready for calibration use. Pre-mixed liquid buffers are 
convenient to use, but have a fairly limited shelf life. Powdered buffer capsules are generally superior 
for long-term storage, and also enjoy the advantage of occupying less storage space in their dry state 
than a liquid buffer solution. The following photograph shows a few 7.00 pH (+/- 0.02 pH) buffer 
capsules ready to be mixed with water to form a usable buffer solution: 

2 For example, a solution with a pH value of 4.7 has a concentration of 10 47 moles of active hydrogen ions per 
liter. For more information on "moles" and solution concentration, see section 2.3, beginning on page 63. 



After preparing the buffer solution in a cup, the pH probe is inserted into the buffer solution and 
given time to stabilize. One stabilized, the pH instrument may be adjusted to register the proper 
pH value. Buffer solutions should not be exposed to ambient air for any longer than necessary 
(especially alkaline buffers such as 10.0 pH) due to contamination 3 . Pre-mixed liquid buffer storage 
containers should be capped immediately after pouring into working cups. Used buffer solution 
should be discarded rather than re-used at a later date. 

Analyzers designed to measure the concentration of certain gases in air must be calibrated 
in a similar manner. Oxygen analyzers, for example, used to measure the concentration of free 
oxygen in the exhaust gases of furnaces, engines, and other combustion processes must be calibrated 
against known standards of oxygen concentration. An oxygen analyzer designed to measure oxygen 
concentration over a range of ambient (20.9% oxygen) to 0% oxygen may be calibrated with ambient 
air as one of the standard values 4 , and a sample of pure nitrogen gas (containing 0% oxygen) as the 

3 Carbon dioxide gas in ambient air will cause carbonic acid to form in an aqueous solution. This has an especially 
rapid effect on high-pH (alkaline) buffers. 

4 It is assumed that the concentration of oxygen in ambient air is a stable enough quantity to serve as a calibration 
standard for most industrial applications. It is certainly an accessible standard! 


other standard value. An oxygen analyzer intended for the measurement of oxygen concentrations in 
excess of ambient air would require a different standard, most likely a sample of 100% pure oxygen, 
as a calibration reference. 

An analyzer designed to measure the concentration of hydrogen sulfide (H 2 S), a toxic gas 
produced by anaerobic bacterial decomposition of organic matter, will require a sample of gas with 
a precisely known concentration of hydrogen sulfide mixed in it as a calibration reference. A typical 
reference gas concentration might be 25 or 50 parts per million (ppm). Gas mixtures with such 
precise concentration values as this may be purchased from chemical laboratories for the purpose of 
calibrating concentration analyzers, and are often referred to as span gases because they are used 
to set the span of analyzer instruments. 

Analytical instruments are generally subject to greater drifting over time than instruments 
that measure incidental quantities such as pressure, level, temperature, or flow rate. It is 
not uncommon for instrument technicians to be tasked with daily calibration checks of certain 
instruments responsible for monitoring atmospheric or water emissions at industrial facilities. For 
this reason, it is often practical to equip such critical analyzers with self- calibration systems. A 
self-calibration system is a system of solenoid (electrically controlled on-off ) valves and reference gas 
bottles set up in such a way that a computer is able to switch the analyzer off-line and subject it to 
standard reference gases on a regular schedule to check calibration. Many analyzers are programmed 
to automatically calibrate themselves against these reference gases, thus eliminating tedious work 
for the instrument technician. A typical self-calibration system for a gas analyzer might look like 



> Output signal 

> Alarm signal 

The gas analyzer is equipped with its own auto-calibration controls and programming, allowing it 
to periodically shut off the process sample and switch to known reference gases for "zero" and "span" 
calibration checks. If these checks indicate excessive drift or any other questionable results, the 
analyzer has the ability to flag a maintenance alarm to alert an instrument technician to a potential 
problem that may require servicing. This sort of self-calibration and self-diagnostic capability saves 
the instrument technician from having to spend substantial time running manual calibration checks, 
yet alerts the technician if anything is in need of actual repair. Barring any component failures within 
this system, the only maintenance this system will need is periodic replacement of the calibration 
gas bottles. 


Calibration: Philosophy In Practice, Second Edition, Fluke Corporation, Everett, WA, 1994. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Chapter 12 

Continuous pressure measurement 

In many ways, pressure is the primary variable for a wide range of process measurements. Many 
types of industrial measurements are actually inferred from pressure, such as: 

• Flow (measuring the pressure dropped across a restriction) 

• Liquid level (measuring the pressure created by a vertical liquid column) 

• Liquid density (measuring the pressure difference across a fixed-height liquid column) 

• Weight (hydraulic load cell) 

Even temperature may be inferred from pressure measurement, as in the case of a fluid-filled 
chamber where fluid pressure and fluid temperature are directly related. As such, pressure is a very 
important quantity to measure, and measure accurately. This section describes different technologies 
for the measurement of pressure. 




12.1 Manometers 

A very simple device used to measure pressure is the manometer: a fluid-filled tube where an applied 
gas pressure causes the fluid height to shift proportionately. This is why pressure is often measured 
in units of liquid height (e.g. inches of water, inches of mercury). As you can see, a manometer is 
fundamentally an instrument of differential pressure measurement, indicating the difference between 
two pressures by a shift in liquid column height: 

U-tube manometer 



* * 

tube allows 

liquid columns 
to be seen 




Liquid column height in a manometer should always be interpreted at the centerline of the liquid 
column, regardless of the shape of the liquid's meniscus (the curved air/liquid interface): 




Read here 

Read here 

Manometers come in a variety of forms, the most common being the U-tube, well (sometimes 
called a cistern), raised well, and inclined: , 

U-tube manometer 



"Well" manometer 



1 Well 1 




"Raised well" manometer 

"Raised-well" inclined manometer input 


| Well | 

U-tube manometers are very inexpensive, and are generally made from clear plastic (see the left- 
hand photo). Cistern-style manometers are the norm for calibration bench work, and are typically 
constructed from metal cisterns and glass tubes (see the right-hand photo): 



Inclined manometers are used to measure very low pressures, owing to their exceptional sensitivity 
(note the fractional scale for inches of water column in the following photograph, extending from 
to 1.5 inches on the scale, reading left to right): 



Note that venting one side of a manometer is standard practice when using is as a gauge pressure 
indicator (responding to pressure in excess of atmospheric) . Both pressure ports will be used if the 
manometer is applied to the measurement of differential pressure, just as in the case of the U-tube 
manometer first shown in this section. Absolute pressure may also be measured by a manometer, if 
one of the pressure ports connects to a sealed vacuum chamber. This is how a mercury barometer 
is constructed for the measurement of absolute ambient air pressure: by sealing off one side of a 
manometer and removing all the air in that side, so that the applied (atmospheric) pressure is always 
compared against a vacuum. 

Manometers incorporating a "well" have the advantage of single-point reading: one need only 
compare the height of one liquid column, not the difference in height between two liquid columns. 
The cross-sectional area of the liquid column in the well is so much greater than that within the 
transparent manometer tube that the change in height within the well is usually negligible. In cases 
where the difference is significant, the spacing between divisions on the manometer scale may be 
skewed to compensate 1 . 

Inclined manometers enjoy the advantage of increased sensitivity. Since manometers 
fundamentally operate on the principle of pressure balanced by liquid height, and this liquid height is 
always measured parallel to the line of gravitational pull (perfectly vertical), inclining the manometer 
tube means that liquid must travel further along the tube to generate the same change in (purely) 
vertical height than it would in a vertical manometer tube. Thus, an inclined manometer tube causes 
an amplification in liquid motion for a given amount of pressure change, allowing measurements of 
greater resolution. 

1 If you are having difficulty understanding this concept, imagine a simple U-tube manometer where one of the tubes 
is opaque, and therefore one of the two liquid columns cannot be seen. In order to be able to measure pressure just 
by looking at one liquid column height, we would have to make a custom scale where every inch of height registered 
as two inches of water column pressure, because for each inch of height change in the liquid column we can see, the 
liquid column we can't see also changes by an inch. A scale custom-made for a well-type manometer is just the same 
concept, only without such dramatic skewing of scales. 


12.2 Mechanical pressure elements 


Mechanical pressure-sensing elements include the bellows, the diaphragm, and the bourdon tube. Each 
of these devices converts a fluid pressure into a force. If unrestrained, the natural elastic properties 
of the element will produce a motion proportional to the applied pressure. 








Applied pressure 



Bellows resemble an accordion constructed from metal instead of fabric. Increasing pressure 
inside a bellows unit causes it to elongate. A diaphragm is nothing more than a thin disk of material 
which bows outward under the influence of a fluid pressure. Many diaphragms are constructed from 
metal, which gives them spring-like qualities. Some diaphragms are intentionally constructed out 
of materials with little strength, such that there is negligible spring effect. These are called slack 
diaphragms, and they are used in conjunction with external mechanisms that produce the necessary 
restraining force to prevent damage from applied pressure. Bourdon tubes are made of spring-like 
metal alloys bent into a circular shape. Under the influence of internal pressure, a bourdon tube 
"tries" to straighten out into its original shape before being bent at the time of manufacture. 

Most pressure gauges use a bourdon tube as their pressure-sensing element. Most pressure 
transmitters use a diaphragm as their pressure-sensing element. Bourdon tubes may be made in 
spiral or helical forms for greater motion (and therefore greater gauge resolution). A typical C-tube 
bourdon tube pressure gauge mechanism is shown in the following illustration: 




Pressure gauge 


A photograph of a C-tube pressure gauge mechanism reveals the physical construction of such a 
pressure gauge: 



It should be noted that bellows, diaphragms, and bourdon tubes alike may all be used to measure 
differential and/or absolute pressure in addition to gauge pressure. All that is needed for these other 
functionalities is to subject the other side of each pressure-sensing element to either another applied 
pressure (in the case of differential measurement) or to a vacuum chamber (in the case of absolute 
pressure measurement): 



Differential pressure sensing mechanisms Applied pressure 

Applied pressure 

Applied pressure 


Applied pressure 



The challenge in doing this, of course, is how to extract the mechanical motion of the pressure- 
sensing element to an external mechanism (such as a pointer) while maintaining a good pressure 
seal. In gauge pressure mechanisms, this is no problem because one side of the pressure-sensing 
element must be exposed to atmospheric pressure anyway, and so that side is always available for 
mechanical connection. 



12.3 Electrical pressure elements 

Several different technologies exist for the conversion of fluid pressure into an electrical signal 
response. These technologies form the basis of electronic pressure transmitters: devices designed 
to measure fluid pressure and transmit that information via electrical signals such as the 4-20 mA 
analog standard, or in digital form such as HART or FOUNDATION Fieldbus. 

A brief survey of electronic pressure transmitters in contemporary 2 use reveals a diverse 
representation of electrical pressure-sensing elements: 

Manufact ur er 


Pressure sensor technology 

ABB /Bailey 


Differential reluctance 

ABB /Bailey 


Piezoresistive (strain gauge) 



Piezoresistive (strain gauge) 



Piezoresistive (strain gauge) 



Differential capacitance 



Differential capacitance 



Differential capacitance 


EJX series 

Mechanical resonance 

2 As of this writing, 2008. 



12.3.1 Piezoresistive (strain gauge) sensors 

Piezoresistive means "pressure-sensitive resistance," or a resistance that changes value with applied 
pressure. The strain gauge is a classic example of a piezoresistive element: 

Test specimen 

As the test specimen is stretched or compressed by the application of force, the conductors of 
the strain gauge are similarly deformed. Electrical resistance of any conductor is proportional to the 
ratio of length over cross-sectional area (R oc 4), which means that tensile deformation (stretching) 
will increase electrical resistance by simultaneously increasing length and decreasing cross-sectional 
area while compressive deformation (squishing) will decrease electrical resistance by simultaneously 
decreasing length and increasing cross-sectional area. 

Attaching a strain gauge to a diaphragm results in a device that changes resistance with applied 
pressure. Pressure forces the diaphragm to deform, which in turn causes the strain gauge to change 
resistance. By measuring this change in resistance, we can infer the amount of pressure applied to 
the diaphragm. 

The classic strain gauge system represented in the previous illustration is made of metal (both 
the test specimen and the strain gauge itself). Within its elastic limits, many metals exhibit good 
spring characteristics. Metals, however, are subject to fatigue over repeated cycles of strain (tension 
and compression), and they will begin to "flow" if strained beyond their elastic limit. This is a 
common source of error in metallic piezoresistive pressure instruments: if overpressured, they tend 
to lose accuracy due to damage of the spring and strain gauge elements. 3 

Modern manufacturing techniques have made possible the construction of strain gauges made 
of silicon instead of metal. Silicon exhibits very linear spring characteristics over its narrow range 
of motion, and a high resistance to fatigue. When a silicon strain gauge is over-stressed, it fails 

3 For a simple demonstration of metal fatigue and metal "flow," simply take a metal paper clip and repeatedly bend 
it back and forth until you feel the metal wire weaken. Gentle force applied to the paper clip will cause it to deform 
in such a way that it returns to its original shape when the force is removed. Greater force, however, will exceed the 
paper clip's elastic limit, causing permanent deformation and also altering the spring characteristics of the clip. 



completely rather than "flows" as is the case with metal strain gauges. This is generally considered 
a better result, as it clearly indicates the need for sensor replacement (whereas a metallic strain 
sensor may give the false impression of continued function after an over-stress event). 

Thus, most modern piezoresistive-based pressure instruments use silicon strain gauge elements 
to sense deformation of a diaphragm due to applied fluid pressure. A simplified illustration of a 
diaphragm / strain gauge pressure sensor is shown here: 


Strain gauge 



In some designs, a single silicon wafer serves as both the diaphragm and the strain gauge so 
as to fully exploit the excellent mechanical properties of silicon (high linearity and low fatigue). 
However, silicon is not chemically compatible with many process fluids, and so pressure must be 
transferred to the silicon diaphragm/sensor via a non-reactive fill fluid (commonly a silicone-based 
or fluorocarbon-based liquid). A metal isolating diaphragm transfers process fluid pressure to the 
fill fluid. Another simplified illustration shows how this works: 

Silicon diaphragm/ 
strain gauge 


Fill fuid 


Metal isolating diaphragm 

Rigid housing 


The isolating diaphragm is designed to be much more flexible (less rigid) than the silicon 
diaphragm, because its purpose is to seamlessly transfer fluid pressure from the process fluid to 



the fill fluid, not to act as a spring element. In this way, the silicon sensor experiences the same 
pressure that it would if it were directly exposed to the process fluid, without having to contact the 
process fluid. 

An example of a pressure instrument utilizing a silicon strain gauge element is the Foxboro model 
IDP10 differential pressure transmitter, shown in the following photograph: 



12.3.2 Differential capacitance sensors 

Another common electrical pressure sensor design works on the principle of differential capacitance. 
In this design, the sensing element is a taut metal diaphragm located equidistant between two 
stationary metal surfaces, forming a complementary pair of capacitances. An electrically insulating 
fill fluid (usually a liquid silicone compound) transfers motion from the isolating diaphragms to the 
sensing diaphragm, and also doubles as an effective dielectric for the two capacitors: 

Output terminals 

Solid insulation 




fill fluid 



Any difference of pressure across the cell will cause the diaphragm to flex in the direction of least 
pressure. Since capacitance between conductors is inversely proportional to the distance separating 
them, this causes capacitance on the low-pressure side to increase and capacitance on the high- 
pressure side to decrease: 



Output terminals 

Solid insulation 






A capacitance detector circuit connected to this cell uses a high-frequency AC excitation signal 
to measure the different in capacitance between the two halves, translating that into a DC signal 
which ultimately becomes the signal output by the instrument representing pressure. 

These pressure sensors are highly accurate, stable, and rugged. The solid frame bounds the 
motion of the two isolating diaphragms such that the sensing diaphragm cannot move past its 
elastic limit. This gives the differential capacitance excellent resistance to overpressure damage. 

A classic example of a pressure instrument based on the differential capacitance sensor is the 
Rosemount model 1151 differential pressure transmitter, shown in assembled form in the following 



By removing four bolts from the transmitter, we are able to remove two flanges from the pressure 
capsule, exposing the isolating diaphragms to plain view: 

A close-up photograph shows the construction of one of the isolating diaphragms, which unlike 



the sensing diaphragm is designed to be very flexible. The concentric corrugations in the metal 
of the diaphragm allow it to easily flex with applied pressure, transmitting process fluid pressure 
through the silicone fill fluid to the taut sensing diaphragm inside the differential capacitance cell: 

The differential capacitance sensor inherently measures differences in pressure applied between 
its two sides. In keeping with this functionality, this pressure instrument has two threaded ports 
into which fluid pressure may be applied. A later section in this chapter will elaborate on the utility 
of differential pressure transmitters (section 12.5 beginning on page 316). 

All the electronic circuitry necessary for converting the sensor's differential capacitance into an 
electronic signal representing pressure is housed in the blue-colored structure above the capsule and 

A more modern realization of the differential capacitance pressure-sensing principle is the 
Rosemount model 3051 differential pressure transmitter: 



Just like the older model, this instrument has two ports through which fluid pressure may be 
applied to the sensor. The sensor, in turn, responds only to the difference in pressure between the 

The differential capacitance sensor construction is more complex in this particular pressure 
instrument, with the plane of the sensing diaphragm lying perpendicular to the plane of the two 
isolating diaphragms. This "coplanar" design is far more compact than the older style of sensor, 
with general engineering advances providing much improved resolution and accuracy. 



12.3.3 Resonant element sensors 

As any guitarist, violinist, or other stringed-instrument musician can tell you, the natural frequency 
of a tensed string increases with tension. This, in fact, is how stringed instruments are tuned: the 
tension on each string is precisely adjusted to achieve the desired resonant frequency. 

Mathematically, the resonant frequency of a string may be described by the following formula: 


/ = Fundamental resonant frequency of string (Hertz) 

L = String length (meters) 

Ft = String tension (newtons) 

/i = Unit mass of string (kilograms per meter) 

It stands to reason, then, that a string may serve as a force sensor. All that is needed to complete 
the sensor is an oscillator circuit to keep the string vibrating at its resonant frequency, and that 
frequency becomes an indication of tension (force) . If the force stems from pressure applied to some 
sensing element such as a bellows or diaphragm, the string's resonant frequency will indicate fluid 
pressure. A proof-of-concept device based on this principle might look like this: 

Applied pressure 

The Foxboro company pioneered this concept in an early resonant wire design of pressure 
transmitter. Later, the Yokogawa corporation of Japan applied the concept to a pair of micro- 
machined 4 silicon resonator structures, which became the basis for their successful line of "DPharp" 
pressure transmitters. A photograph of a Yokogawa model EJA110 pressure transmitter with this 
technology is seen here: 

4 This is an example of a micro-electro-mechanical system, or MEMS. 



Even when disassembled, the transmitter does not look much different from the more common 
differential capacitance sensor design. Process pressure enters through ports in two flanges, presses 
against a pair of isolating diaphragms, transferring motion to the sensing diaphragm where the 
resonant elements change frequency with diaphragm strain: 



The important design differences are hidden from view, inside the sensing capsule. Functionally, 
though, this transmitter is much the same as its differential-capacitance cousin. 

An interesting advantage of the resonant element pressure sensor is that the sensor signal is very 
easy to digitize. The vibration of each resonant element is sensed by the electronics package as an AC 
frequency. Any frequency signal may be easily "counted" over a given span of time and converted to 
a binary digital representation. Quartz crystal electronic oscillators are extremely precise, providing 
the stable frequency reference necessary for comparison in any frequency-based instrument. 

In the Yokogawa "DPharp" design, the two resonant elements oscillate at a nominal frequency 
of 90 kHz. As the sensing diaphragm deforms with applied differential pressure, one resonator 
experiences tension while the other experiences compression, causing the frequency of the former to 
shift up and the latter to shift down (as much as +/- 20 kHz). The signal conditioning electronics 
inside the transmitter measures this difference in resonator frequency to infer applied pressure. 


12.3.4 Mechanical adaptations 

Most modern electronic pressure sensors convert very small diaphragm motions into electrical signals 
through the use of sensitive motion-sensing techniques (strain gauge sensors, differential capacitance 
cells, etc.). Diaphragms made from elastic materials behave as springs, but circular diaphragms 
exhibit very nonlinear behavior when significantly stretched unlike classic spring designs such as 
coil and leaf springs which exhibit linear behavior over a wide range of motion. Therefore, in 
order to yield a linear response to pressure, a diaphragm-based pressure sensor must be designed in 
such a way that the diaphragm stretches very little over the normal range of operation. Limiting 
the displacement of a diaphragm necessitates highly sensitive motion-detection techniques such as 
strain gauge sensors, differential capacitance cells, and mechanical resonance sensors to convert that 
diaphragm's very slight motion into an electronic signal. 

An alternative approach to electronic pressure measurement is to use mechanical pressure- 
sensing elements with more linear pressure-displacement characteristics - such as bourdon tubes 
and spring-loaded bellows - and then detect the large-scale motion of the pressure element using 
a less-sophisticated electrical motion-sensing device such as a potentiometer, LVDT, or Hall Effect 
sensor. In other words, we take the sort of mechanism commonly found in a direct-reading pressure 
gauge and attach it to a potentiometer (or similar device) to derive an electrical signal from the 
pressure measurement. 

This alternative approach is undeniably simpler and less expensive to manufacture than the 
more sophisticated approaches used with diaphragm-based pressure instruments, but is prone to 
greater inaccuracies. Even bourdon tubes and bellows are not perfectly linear spring elements, and 
the substantial motions involved with using such pressure elements introduces the possibility of 
hysteresis errors (where the instrument does not respond accurately during reversals of pressure, 
where the mechanism changes direction of motion) due to mechanism friction, and deadband errors 
due to backlash (looseness) in mechanical connections. 

You are likely to encounter this sort of pressure instrument design in direct-reading gauges 
equipped with electronic transmitting capability. An instrument manufacturer will take a proven 
product line of pressure gauge and add a motion-sensing device to it that generates an electric 
signal proportional to mechanical movement inside the gauge, resulting in an inexpensive pressure 
transmitter that happens to double as a direct-reading pressure gauge. 


12 A Force-balance pressure transmitters 

An important legacy technology for all kinds of continuous measurement is the self-balancing system. 
A "self-balance" system continuously balances an adjustable quantity against a sensed quantity, the 
adjustable quantity becoming an indication of the sensed quantity once balance is achieved. A 
common manual-balance system is the type of scale used in laboratories to measure mass: 


Known masses 

Unknown mass 

Here, the unknown mass is the sensed quantity, and the known masses are the adjustable quantity. 
A human lab technician applies as many masses to the left-hand side of the scale as needed to achieve 
balance, then counts up the sum total of those masses to determine the quantity of the unknown 

Such a system is perfectly linear, which is why these balance scales are popularly used for scientific 
work. The scale mechanism itself is the very model of simplicity, and the only thing the pointer 
needs to accurately sense is a condition of balance (equality between masses) . 

If the task of balancing is given to an automatic mechanism, the adjustable quantity will 
continuously change and adapt as needed to balance the sensed quantity, thereby becoming a 
representation of that sensed quantity. In the case of pressure instruments, pressure is easily 
converted into force by acting on the surface area of a sensing element such as a diaphragm or a 
bellows. A balancing force may be generated to exactly cancel the process pressure's force, making a 
force-balance pressure instrument. Like the laboratory balance scale, an industrial instrument built 
on the principle of balancing a sensed quantity with an adjustable quantity will be inherently linear, 
which is a tremendous advantage for measurement purposes. 

Here, we see a diagram of a force-balance pneumatic pressure transmitter 5 , balancing a sensed 
differential pressure with an adjustable air pressure which becomes a pneumatic output signal: 

3 Based on the design of Foxboro's popular model 13A pneumatic "DP cell" differential pressure transmitter. 




High pressure 

input ► 



Diaphragm seal 


Zero screw 

Low pressure 
< input 

Differential pressure is sensed by a liquid-filled diaphragm "capsule," which transmits force to 
a "force bar." If the force bar moves out of position due to this applied force, a highly sensitive 
"baffle" and "nozzle" mechanism senses it and causes a pneumatic amplifier (called a "relay") to 
send a different amount of air pressure to a bellows unit. The bellows presses against the "range 
bar" which pivots to counter-act the initial motion of the force bar. When the system returns to 
equilibrium, the air pressure inside the bellows will be a direct, linear representation of the process 
fluid pressure applied to the diaphragm capsule. 

With minor modifications to the design of this pressure transmitter 6 , we may convert it from 
pneumatic to electronic force-balancing: 

3 Very loosely based on the design of Foxboro's popular E13 electronic "DP cell" differential pressure transmitter. 




Diaphragm seal 

High pressure 


w Force 
(| (adjustable) 

Range wheel 

output signal 

Zero screw 

Low pressure 
— input 

Differential pressure is sensed by the same type of liquid-filled diaphragm capsule, which transmits 
force to the force bar. If the force bar moves out of position due to this applied force, a highly sensitive 
electromagnetic sensor detects it and causes an electronic amplifier to send a different amount of 
electric current to a force coil. The force coil presses against the range bar which pivots to counter- 
act the initial motion of the force bar. When the system returns to equilibrium, the milliampere 
current through the force coil will be a direct, linear representation of the process fluid pressure 
applied to the diaphragm capsule. 

A distinct advantage of force-balance pressure instruments (besides their inherent linearity) is 
the constraining of sensing element motion. Unlike a modern diaphragm-based pressure transmitter 
which relies on the spring characteristics of the diaphragm to convert pressure into force and then 
into motion (displacement) which is sensed and converted into an electronic signal, a force-balance 
transmitter works best when the diaphragm is slack and has no spring characteristics at all. Balance 
with the force of the process fluid pressure is achieved by the application of either an adjustable 
air pressure or an adjustable electric current, not by the natural tensing of a spring element. This 
makes a force-balance instrument far less susceptible to errors due to metal fatigue or any other 


degradation of spring characteristics. 

Unfortunately, force-balance instruments have significant disadvantages as well. Force-balance 
mechanisms tend to be bulky 7 , and they translate external vibration into inertial force which adds 
"noise" to the output signal. Also, the amount of electrical power necessary to provide adequate 
balancing force in an electronic force-balance transmitter is such that it is nearly impossible to limit 
below the level necessary to ensure intrinsic safety (protection against the accidental ignition of 
explosive atmospheres by limiting the amount of energy the instrument could possibly discharge 
into a spark). 

7 One instrument technician I encountered referred to the Foxboro E13 differential pressure transmitter as "pig 
iron" after having to hoist it by hand to the top of a distillation tower. 


12.5 Differential pressure transmitters 

One of the most common, and most useful, pressure measuring instruments in industry is the 
differential pressure transmitter. This device senses the difference in pressure between two ports and 
outputs a signal representing that pressure in relation to a calibrated range. Differential pressure 
transmitters may be based on any of the previously discussed pressure-sensing technologies, so this 
section discusses practical application rather than internal workings. 

Differential pressure transmitters look something like this: 

Pneumatic DP 


* Air signal out 
Air supply 

Diaphragm capsule 

Electronic DP 

I Wires 


Diaphragm capsule 

Two models of electronic differential pressure transmitter are shown here, the Rosemount model 
1151 (left) and model 3051 (right): 

Two more models of electronic differential pressure transmitter are shown in the next photograph, 
the Yokogawa EJA110 (left) and the Foxboro IDP10 (right): 



Regardless of make or model, every differential pressure ("DP", "d/p", or AP) 8 transmitter has 
two pressure ports to sense different process fluid pressures. One of these ports is labeled "high" 
and the other is labeled "low" . This labeling does not necessarily mean that the "high" port must 
always be at a greater pressure than the "low" port. What these labels represent is the effect that 
a pressure at that point will have on the output signal. 

The concept of differential pressure instrument port labeling is very similar to the "inverting" 
and "noninverting" labels applied to operational amplifier input terminals: 


The "+" and "-" symbols do not imply polarity of the input voltage(s). It is not as though the 
"+" input must be more positive than the "-" input. These symbols merely represent the different 
effects on the output signal that each input has. An increasing voltage applied to the "+" input 
drives the op-amp's output positive, while an increasing voltage applied to the "-" input drives the 
op-amp's output negative. In a similar manner, an increasing pressure applied to the "high" port 
of a DP transmitter will drive the output signal to a greater level (up), while an increasing pressure 
applied to the "low" port of a DP transmitter will drive the output signal to a lesser level (down) : 

8 As far as I have been able to determine, the labels "D/P" and "DP cell" were originally trademarks of the Foxboro 
Company. Those particular transmitter models became so popular that the term "DP cell" came to be applied to 
nearly all makes and models of differential pressure transmitter, much like the trademark "Vise-Grip" is often used 
to describe any self-locking pliers, or "Band-Aid" is often used to describe any form of self-adhesive bandage. 



Pressure here 
drives output 
toward 20 mA 


4-20 mA signal 

Pressure here 

- drives output 

toward 4 mA 



We can use metal or plastic tubes (or pipes) to connect one or more ports of a pressure transmitter 
to points in a process. These tubes are commonly called impulse lines, or gauge lines, or sensing 
lines 9 . This is equivalent to the test wires used to connect a voltmeter to points in a circuit for 
measuring voltage. Typically, these tubes are connected to the transmitter and to the process by 
means of compression fittings which allow for relatively easy disconnection and reconnection of tubes. 

The combination of two differential pressure ports makes the DP transmitter very versatile as 
a pressure-measuring device. We may use the DP transmitter to measure an actual difference of 
pressure across a fluid device such as a filter. Here, the amount of differential pressure across the 
filter represents how clogged the filter is: 

3 Also called impulse tubes, gauge tubes, or sensing tubes. 



Impulse line 



Impulse line 

—CD £1 III CD CD. 

w — <g — <g — <b — <s — W — W 




-Led cd cd cd cd cd cd. 




Note how the high side of the DP transmitter connects to the upstream side of the filter, and 
the low side of the transmitter to the downstream side of the filter. This way, increased filter 
clogging will result in an increased transmitter output. Since the transmitter's internal pressure- 
sensing diaphragm only responds to differences in pressure between the "high" and "low" ports, the 
pressure in the filter and pipe relative to the atmosphere is completely irrelevant to the transmitter's 
output signal. The filter could be operating at a pressure of 10 PSI or 10,000 PSI: the only thing 
the DP transmitter measures is the pressure drop across the filter. If the upstream side is at 10 PSI 
and the downstream side is at 9 PSI, the differential pressure will be 1 PSI (sometimes labeled as 
PSID, "D" for differential) . If the upstream pressure is 10,000 PSI and the downstream pressure 
is 9,999 PSI, the DP transmitter will still see a differential pressure of just 1 PSID. Likewise, the 
technician calibrating the DP transmitter on the workbench could use a precise air pressure of just 
1 PSI (applied to the "high" port, with the "low" port vented to atmosphere) to simulate either 
of these real-world conditions. The DP transmitter simply cannot tell the difference between these 
three scenarios, nor should it be able to tell the difference if its purpose is to exclusively measure 
differential pressure. 

In the world of electronics, we refer to the ability of a differential voltage sensor (such as an 



operational amplifier) to sense small differences in voltage while ignoring large potentials measured 
with reference to ground by the phrase common-mode rejection. An ideal operational amplifier 
completely ignores the amount of voltage common to both input terminals, responding only to the 
difference in voltage between those terminals. This is precisely what a well-designed differential 
pressure instrument does, except with fluid pressure instead of electrical voltage. A differential 
pressure instrument all but ignores gauge pressure common to both ports, while responding only to 
differences in pressure between those two ports. 

A vivid example of this may be inferred from the nameplate of a Foxboro model 13A differential 
pressure transmitter, shown in this photograph: 

This nameplate tells us that the transmitter has a calibrated differential pressure range of 50" 
H2O (50 inches water column, which is only about 1.8 PSI). However, the nameplate also tells us 
that the transmitter has a maximum working pressure (MWP) of 1500 PSI. "Working pressure" 
refers to the amount of gauge pressure common to each port, not the differential pressure between 
ports. Taking these figures at face value means this transmitter will register zero (no differential 
pressure) even if the gauge pressure applied equally to both ports is a full 1500 PSI! In other words, 
this differential pressure transmitter will reject up to 1500 PSI of gauge pressure, and respond only 
to small differences in pressure between the ports (1.8 PSI differential being enough to stimulate the 
transmitter to full scale output). 


12.6 Pressure sensor accessories 

Multiple accessories exist for pressure-sensing devices to function optimally in challenging process 
environments. Sometimes, we must use special accessories to protect the pressure instrument against 
hazards of certain process fluids. One such hazard is pressure pulsation, for example at the discharge 
of a piston-type (positive-displacement) high-pressure pump. Pulsating pressure can quickly damage 
mechanical sensors such as bourdon tubes, either by wear of the mechanism transferring pressure 
element motion to an indicating needle, and/or fatigue of the metal element itself. 



12.6.1 Valve manifolds 

An important accessory to the differential pressure transmitter is the three-valve manifold. This 
device incorporates three manual valves to isolate and equalize pressure from the process to the 
transmitter, for maintenance and calibration purposes. 

The following illustration shows the three valves comprising a three-valve manifold (within the 
dotted-line box) , as well as a fourth valve called a "bleed" valve used to vent trapped fluid pressure 
to atmosphere: 

Block valve 

Bleed valve 


Equalizing valve 
Block valve 

Impulse lines to process . . . 

While this illustration shows the three valves as separate devices, connected together and to 
the transmitter by tubing, three-valve manifolds are more commonly manufactured as monolithic 
devices: the three valves cast together into one block of metal, attaching to the pressure transmitter 
by way of a flanged face with O-ring seals. Bleed valves are most commonly found as separate 
devices threaded into one or more of the ports on the transmitter's diaphragm chambers. 

The following photograph shows a three-valve manifold bolted to a Honeywell model ST3000 
differential pressure transmitter. A bleed valve fitting may be seen inserted into the upper port on 
the nearest diaphragm capsule flange: 



In normal operation, the two block valves are left open so that process fluid pressure may reach 
the transmitter. The equalizing valve is left tightly shut so that no fluid can pass between the "high" 
and "low" pressure sides. To isolate the transmitter from the process for maintenance, one must first 
close the block valves, then open the equalizing valve to ensure the transmitter "sees" no differential 
pressure. The "bleed" valve is opened at the very last step to relieve pent-up fluid pressure within 
the manifold and transmitter chambers: 

Normal operation 

Removed from service 



| Shut 




A variation on this theme is the five-valve manifold, shown in this illustration: 



Equalizing valve 

Block valve 

Bleed valve 

Equalizing valve 

Block valve 

To process 

To process 

To atmosphere 
(or safe location elsewhere) 

Manifold valve positions for normal operation and maintenance are as follows: 

Normal operation Removed from service 


/ — *, 


^ — * 

Open T Open 

LrTl °P en - -* * ■ 



Shut 1 

It is critically important that the equalizing valve (s) never be open while both block valves 
are open! Doing so will allow process fluid to flow through the equalizing valve(s) from the high- 
pressure side of the process to the low-pressure side of the process. If the impulse tubes connecting 
the manifold to the process are intentionally filled with a fill fluid (such as glycerin, to displace 
process water from entering the impulse tubes; or water in a steam system), this fill fluid will be 
lost. Also, if the process fluid is dangerously hot or radioactive, a combination of open equalizing 
and block valves will let that dangerous fluid reach the transmitter and manifold, possibly causing 
damage or creating a personal hazard. Speaking from personal experience, I once made this mistake 
on a differential pressure transmitter connected to a steam system, causing hot steam to flow through 
the manifold and overheat the equalizing valve so that it seized open and could not be shut again! 
The only way I was able to stop the flow of hot steam through the manifold was to locate and shut 



a sliding-gate hand valve between the impulse tube and the process pipe. Fortunately, this cast iron 
valve was not damaged by the heat and was still able to shut off the flow. 

Pressure transmitter valve manifolds also come in single block-and-bleed configurations, for gauge 
pressure applications. Here, the "low" pressure port of the transmitter is vented to atmosphere, with 
only the "high" pressure port connected to the impulse line: 

Block valve 

Bleed valve 

Impulse line to process . . . 

The following photograph shows a bank of eight pressure transmitters, seven out of the eight being 
equipped with a single block-and-bleed manifold. The eighth transmitter (bottom row, second-from 
left) sports a 5-valve manifold: 



12.6.2 Bleed fittings 

Before removing a pressure transmitter from live service, the technician must "bleed" stored fluid 
pressure to atmosphere in order to achieve a zero energy state prior to disconnecting the transmitter 
from the impulse lines. Some valve manifolds provide a bleed valve for doing just this, but 
many do not 10 . An inexpensive and common accessory for pressure-sensing instruments (especially 
transmitters) is the bleed valve fitting, installed on the instrument as a discrete device. The most 
common bleed fitting is equipped with 1/4 inch male NPT pipe threads, for installation into one 
of the 1/4 inch NPT threaded pipe holes typically provided on pressure transmitter flanges. The 
bleed is operated with a small wrench, loosening a ball-tipped plug off its seat to allow process 
fluid to escape through a small vent hole in the side of the fitting. The following photographs show 
close-up views of a bleed fitting both assembled (left) and with the plug fully extracted from the 
fitting (right). The bleed hole may be clearly seen in both photographs: 

When installed directly on the flanges of a pressure instrument, these bleed valves may be used 
to bleed unwanted fluids from the pressure chambers, for example bleeding air bubbles from an 
instrument intended to sense water pressure, or bleeding condensed water out of an instrument 
intended to sense compressed air pressure. 

The following photographs show bleed fittings installed two different ways on the side of a pressure 
transmitter flange, one way to bleed gas out of a liquid process (located on top) and the other way 
to bleed liquid out of a gas process (located on bottom): 

] The standard 3-valve manifold, for instance, does not provide a bleed valve — only block and equalizing valves. 


12.6.3 Pressure pulsation dampening 

A simple way to mitigate the effects of pulsation on a pressure gauge is to fill the inside of the 
gauge with a viscous liquid such as glycerin or oil. The inherent friction of this fill liquid has a 
"shock-absorber" quality which dampens the gauge mechanism's motion and helps protect against 
damage from pulsations or from external vibration. This method is ineffectual for high-amplitude 
pulsations, though. 

A more sophisticated method for dampening pulsations seen by a pressure instrument is called a 
snubber, and it consists of a fluid restriction placed between with the pressure sensor and the process. 
The simplest example of a snubber is a simple needle valve (an adjustable valve designed for low 
flow rates) placed in a mid-open position, restricting fluid flow in and out of a pressure gauge: 

(&\ Pressure gauge 
Needle valve 

(partially open) 


At first, the placement of a throttling valve between the process and a pressure- measuring 
instrument seems rather strange, because there should not be any continuous flow in or out of 
the gauge for such a valve to throttle! However, a pulsing pressure causes a small amount of 
alternating flow in and out of the pressure instrument, owing to the expansion and contraction of 
the mechanical pressure-sensing element (bellows, diaphragm, or bourdon tube). The needle valve 
provides a restriction for this flow which, when combined with the fluid capacitance of the pressure 
instrument, combine to form a low-pass filter of sorts. By impeding the flow of fluid in and out 
of the pressure instrument, that instrument is prevented from "seeing" the high and low peaks 
of the pulsating pressure. Instead, the instrument registers a much steadier pressure over time. 
An electrical analogy for a pressure snubber is an RC low-pass filter circuit "dampening" voltage 
pulsations from reaching a voltmeter: 



"Pulsing" voltage 
source (DC + AC) 

Voltmeter needle 

vibrates with AC 



"Pulsing" voltage 
source (DC + AC) 

Low-pass filter 

Voltmeter needle 
no longer 

One potential problem with the needle valve idea is that the small orifice inside the valve may 
plug up over time with debris or deposits from dirty process fluid. This, of course, would be bad 
because if that valve were to ever completely plug, the pressure instrument would stop responding 
to any changes in process pressure at all, or perhaps just become too slow in responding to major 

A solution to this problem is to fill the pressure sensor mechanism with a clean liquid (called a fill 
fluid), then transfer pressure from the process fluid to the fill fluid (and then to the pressure-sensing 
element) using a slack diaphragm or some other membrane: 

<^X Pressure gauge 
Needle valve 

(partially open) 

Isolating diaphragm 


In order for the fill fluid and isolating diaphragm to work effectively, there cannot be any gas 
bubbles in the fill fluid - it must be a "solid" hydraulic system from the diaphragm to the sensing 
element. The presence of gas bubbles means that the fill fluid is compressible, which means the 


isolating diaphragm may have to move more than necessary to transfer pressure to the instrument's 
sensing element. This will introduce pressure measurement errors if the isolating diaphragm begins to 
tense from excessive motion (and thereby oppose some process fluid pressure from fully transferring to 
the fill fluid), or hit a "stop" point where it cannot move any further (thereby preventing any further 
transfer of pressure from process fluid to fill fluid) 11 . For this reason, isolating diaphragm systems 
for pressure instruments are usually "packed" with fill fluid at the point and time of manufacture, 
then sealed in such a way that they cannot be opened for any form of maintenance. Consequently, 
any fill fluid leak in such a system immediately ruins it. 

11 This concept will be immediately familiar to anyone who has ever had to "bleed" air bubbles out of an automobile 
brake system. With air bubbles in the system, the brake pedal has a "spongy" feel when depressed, and much 
pedal motion is required to achieve adequate braking force. After bleeding all air out of the brake fluid tubes, the 
pedal motion feels much more "solid" than before, with minimal motion required to achieve adequate braking force. 
Imagine the brake pedal being the isolating diaphragm, and the brake pads being the pressure sensing element inside 
the instrument. If enough gas bubbles exist in the tubes, the brake pedal might stop against the floor when fully 
pressed, preventing full force from ever reaching the brake pads! Likewise, if the isolating diaphragm hits a hard 
motion limit due to gas bubbles in the fill fluid, the sensing element will not experience full process pressure! 



12.6.4 Remote and chemical seals 

Isolating diaphragms have merit even in scenarios where pressure pulsations are not a problem. 
Consider the case of a food-processing system where we must remotely measure pressure inside a 
mixing vessel: 


Pressure gauge 

The presence of the tube connecting the vessel to the pressure gauge poses a hygiene problem. 
Stagnant process fluid (in this case, some liquid food product) inside the tube can support microbial 
growth, which will eventually contaminate the vessel no matter how well or how often the vessel is 
cleaned. Even automated Clean- In- Place and Steam- In- Place (CIP and SIP, respectively) protocols 
where the vessel is chemically purged between batches cannot prevent this problem because the 
cleaning agents never purge the entire length of the tubing (ultimately, to the bourdon tube or other 
sensing element inside the gauge). 

Here, we see a valid application of an isolating diaphragm and fill fluid. If we mount an isolating 
diaphragm to the vessel in such a way that the process fluid directly contacts the diaphragm, 
sealed fill fluid will be the only material inside the tubing carrying that pressure to the instrument. 
Furthermore, the isolating diaphragm will be directly exposed to the vessel interior, and therefore 
cleaned with every CIP cycle. Thus, the problem of microbial contamination is completely avoided: 





Pressure gauge 


Capillary tubing 

(with fill fluid) 

Such systems are often referred to as remote seals, and they are available on a number of different 
pressure instruments including gauges, transmitters, and switches. If the purpose of an isolating 
diaphragm and fill fluid is to protect the sensitive instrument from corrosive or otherwise harsh 
chemicals, it is often referred to as a chemical seal. 

The following photograph shows a pressure gauge equipped with a chemical seal diaphragm. 
Note that the chemical seal on this particular gauge is close-coupled to the gauge, since the only 
goal here is protection of the gauge from harsh process fluids, not the ability to remotely mount the 



A view facing the bottom of the flange reveals the thin metal isolating diaphragm which keeps 
process fluid from entering the gauge mechanism. Only inert fill fluid occupies the space between 
this diaphragm and the gauge's bourdon tube: 



The only difference between this chemical-seal gauge and a remote-seal gauge is that a remote- 
seal gauge uses a length of very small-diameter tubing called capillary tubing to transfer fill fluid 
pressure from the sealing diaphragm to the gauge mechanism. 

Direct-reading gauges are not the only type of pressure instrument that may benefit from having 
remote seals. Electronic pressure transmitters are also manufactured with remote seals for the same 
reasons: protection of the transmitter sensor from harsh process fluid, or prevention of "dead-end" 
tube lengths where organic process fluid would stagnate and harbor microbial growths. The following 
photograph shows a pressure transmitter equipped with a remote sealing diaphragm. The capillary 
tube is protected by a coiled metal ("armor") sheath: 

A close-up view of the sealing diaphragm shows its corrugated design, allowing the metal to 
easily flex and transfer pressure to the fill fluid within the capillary tubing: 

Just like the isolating diaphragms of the pressure-sensing capsule, these remote diaphragms need 
only transfer process fluid pressure to the fill fluid and (ultimately) to the taut sensing diaphragm 
inside the instrument. Therefore, these diaphragms perform their function best if they are designed 



to easily flex. This allows the taut sensing diaphragm to provide the vast majority of the opposing 
force to the fluid pressure, as though it were the only spring element in the fluid system. 

The connection point between the capillary tube and the transmitter's sensor capsule is labeled 
with a warning never to disassemble, since doing so would allow air to enter the filled system (or fill 
fluid to escape from the system) and thereby ruin its accuracy: 

In order for a remote seal system to work, the hydraulic "connection" between the sealing 
diaphragm and the pressure-sensing element must be completely gas-free so that there will be a 
"solid" transfer of motion from one end to the other. 

A potential problem with using remote diaphragms is the hydrostatic pressure generated by the 
fill fluid if the pressure instrument is located far away (vertically) from the process connection point. 
For example, a pressure gauge located far below the vessel it connects to will register a greater 
pressure than what is actually inside the vessel, because the vessel's pressure adds to the hydrostatic 
pressure caused by the liquid in the tubing: 




Pressure = P. 
to 50 PSI 



Tube / 

(with fill fluid) 

P elevation = P8 h = Y h 

Elevation = 4 -86PSI 

reSSUre — rp rocess + "elevation 

|J Pressure gauge 

4.86 to 54.86 PSI 

(calibrated range) 

This pressure may be calculated by the formula pgh or jh where p is the mass density of the fill 
liquid or 7 is the weight density of the fill liquid. For example, a 12 foot capillary tube height filled 
with a fill liquid having a weight density of 58.3 lb/ft 3 will generate an elevation pressure of almost 
700 lb/ft 2 , or 4.86 PSI. If the pressure instrument is located below the process connection point, 
this 4.86 PSI offset must be incorporated into the instrument's calibration range. If we desire this 
pressure instrument to accurately measure a process pressure range of to 50 PSI, we would have 
to calibrate it for an actual range of 4.86 to 54.86 PSI. 

The reverse problem exists where the pressure instrument is located higher than the process 
connection: here the instrument will register a lower pressure than what is actually inside the 
vessel, offset by the amount predicted by the hydrostatic pressure formulae pgh or 7/1. 

In all fairness, this problem is not limited to remote seal systems - even non-isolated systems 
where the tubing is filled with process liquid will exhibit this offset error. However, in filled-capillary 
systems a vertical offset is guaranteed to produce a pressure offset because fill fluids are always liquid, 
and liquids generate pressure in direct proportion to the vertical height of the liquid column (and 
to the density of that liquid) . 

A similar problem unique to isolated-fill pressure instruments is measurement error caused by 


temperature extremes. Suppose the liquid-filled capillary tube of a remote seal pressure instrument 
comes too near a hot steam pipe, furnace, or some other source of high temperature. The expansion 
of the fill fluid may cause the isolation diaphragm to extend to the point where it begins to tense 
and add a pressure to the fill fluid above and beyond that of the process fluid. Cold temperatures 
may wreak havoc with filled capillary tubes as well, if the fill fluid congeals or even freezes such that 
it no longer flows as it should. 

Proper mounting of the instrument and proper selection of the fill fluid 12 will help to avoid such 
problems. All in all, the potential for trouble with remote- and chemical-seal pressure instruments 
is greatly offset by their benefits in the right applications. 

12 Most pressure instrument manufacturers offer a range of fill fluids for different applications. Not only is 
temperature a consideration in the selection of the right fill fluid, but also potential contamination of or reaction 
with the process if the isolating diaphragm ever suffers a leak! 



12.6.5 Filled impulse lines 

An alternate method for isolating a pressure-sensing instrument from direct contact with process 
fluid is to either fill or purge the impulse lines with a harmless fluid. Filling impulse tubes with a 
static fluid works when gravity is able to keep the fill fluid in place, such as in this example of a 
pressure transmitter connected to a water pipe by a glycerin-filled impulse line: 

Water pipe 


Isolation ("block") valve 

Impulse line 

(filled with glycerin which 
is denser than water) 

Pressure transmitter 

i | \ ^ Air supply 

Fill valve 


PV signal 

A reason someone might do this is for freeze protection, since glycerin freezes at a lower 
temperature than water. If the impulse line were filled with process water, it might freeze solid 
in cold weather conditions (the water in the pipe cannot freeze so long as it is forced to flow). The 
greater density of glycerin keeps it placed in the impulse line, below the process water line. A fill 
valve is provided near the transmitter so that a technician may re-fill the impulse line with glycerin 
(using a hand pump) if ever needed. 

As with a remote diaphragm, a filled impulse line will generate its own pressure proportional to 
the height difference between the point of process connection and the pressure-sensing element. If 
the height difference is substantial, the pressure offset resulting from this difference in elevation will 
require compensation by means of an intentional "zero shift" of the pressure instrument when it is 



12.6.6 Purged impulse lines 

Continuous purge of an impulse line is an option when the line is prone to plugging. Consider this 
example, where pressure is measured at the bottom of a sedimentation vessel: 

Pressure transmitter 

r — ^ 1 Air supply 

PV signal 

water supply 

( f s —[ ? Purge 

^ ' ' ' water sur 

/ \ 

Purge valve Check valve 

A continuous flow of clean water enters through a "purge valve" and flows through the impulse 
line, keeping it clear of sediment while still allowing the pressure instrument to sense pressure at 


the bottom of the vessel. A check valve guards against reverse flow through the purge line, in case 
process fluid pressure ever exceeds purge supply pressure. Purged systems are very useful, but a few 
details are necessary to consider before deciding to implement such a strategy: 

• How reliable is the supply of purge fluid? If this stops for any reason, the impulse line may 


Is the purge fluid supply pressure guaranteed to exceed the process pressure at all times, for 
proper direction of purge flow? 

• What options exist for purge fluids that will not adversely react with the process? 

• What options exist for purge fluids that will not contaminate the process? 

• How expensive will it be to maintain this constant flow of purge fluid into the process? 

Also, it is important to limit the flow of purge fluid to a rate that will not create a falsely high 
pressure measurement due to restrictive pressure drop across the length of the impulse line, yet flow 
freely enough to achieve the goal of plug prevention. In many installations, a visual flow indicator 
is installed in the purge line to facilitate optimum purge flow adjustment. Such flow indicators are 
also helpful for troubleshooting, as they will indicate if anything happens to stop the purge flow. 

In the previous example, the purge fluid was clean water. Many options exist for purge fluids 
other than water, though. Gases such as air, nitrogen, or carbon dioxide are often used in purged 
systems, for both gas and liquid process applications. 

Purged impulse lines, just like filled lines and diaphragm-isolated lines, will generate hydrostatic 
pressure with vertical height. If the purge fluid is a liquid, this elevation-dependent pressure may be 
an offset to include in the instrument's calibration. If the purge fluid is a gas (such as air), however, 
any height difference may be ignored because the density of the gas is negligible. 



12.6.7 Heat-traced impulse lines 

If impulse lines are filled with liquid, there may exist a possibility for that liquid to freeze in cold- 
weather conditions. This possibility depends, of course, on the type of liquid filling the impulse lines 
and how cold the weather gets in that geographic location. 

One safeguard against impulse line freezing is to trace the impulse lines with some form of 
active heating medium, steam and electrical being the most common. "Steam tracing" consists of a 
copper tube carrying low-pressure steam, bundled alongside one or more impulse tubes, enclosed in 
a thermally insulating jacket. 

Water pipe 

15PSI 2_ 
steam supply ' 



Isolation ("block") valve 

Steam -traced 
~ impulse tube 

Pressure gauge 

Steam "trap" 


Steam flows through the shutoff valve, through the tube in the insulated bundle, transferring 
heat to the impulse tube as it flows past. Cooled steam condenses into water and collects in the 
steam trap device located at the lowest elevation on the steam trace line. When the water level 
builds up to a certain level inside the trap, a float-operated valve opens to vent the water. This 
allows more steam to flow into the tracing tube, keeping the impulse line continually heated. 

The steam trap naturally acts as a sort of thermostat as well, even though it only senses condensed 
water level and not temperature. The rate at which steam condenses into water depends on how 
cold the impulse tube is. The colder the impulse tube (caused by colder ambient conditions), the 
more heat energy drawn from the steam, and consequently the faster condensation rate of steam 
into water. This means water will accumulate faster in the steam trap, which means it will "blow 
down" more often. More frequent blow-down events means a greater flow rate of steam into the 
tracing tube, which adds more heat to the tubing bundle and raises its temperature. Thus, the 
system is naturally regulating, with its own negative feedback loop to maintain bundle temperature 
at a relatively stable point 13 . 

13 In fact, after you become accustomed to the regular "popping" and "hissing" sounds of steam traps blowing 
down, you can interpret the blow-down frequency as a crude ambient temperature thermometer! Steam traps seldom 


Steam traps are not infallible, being susceptible to freezing (in very cold weather) and sticking 
open (wasting steam by venting it directly to atmosphere). However, they are generally reliable 
devices, capable of adding tremendous amounts of heat to impulse tubing for protection against 

Electrically traced impulse lines are an alternative solution for cold-weather problems. The 
"tracing" used is a twin-wire cable (sometimes called heat tape) that acts as a resistive heater. 
When power is applied, the cable heats up, thus imparting thermal energy to the impulse tubing it 
is bundled with. 

Heat tape may be self-regulating, or controlled with an external thermostat. Self-regulating heat 
tape exhibits an electrical resistance that varies with temperature, automatically self-regulating its 
own temperature without the need for external controls. 

Both steam and electrical heat tracing are used to protect instruments themselves from cold 
weather freezing, not just the impulse lines. In these applications it is important to remember that 
only the liquid-filled portions of the instrument need freeze protection, not the electronics portions! 

blow down during warm weather, but their "popping" is much more regular (one every minute or less) when ambient 
temperatures drop well below the freezing point of water. 



12.6.8 Water traps and pigtail siphons 

Many industrial processes utilize high-pressure steam for direct heating, performing mechanical work, 
combustion control, and as a chemical reactant. Measuring the pressure of steam is important both 
for its end-point use and its generation (in a boiler). One problem with doing this is the relatively 
high temperature of steam at the pressures common in industry, which can cause damage to the 
sensing element of a pressure instrument if directly connected. 

A simple yet effective solution to this problem is to intentionally create a "low" spot in the 
impulse line where condensed steam (water) will accumulate and act as a liquid barrier to prevent 
hot steam from reaching the pressure instrument. The principle is much the same as a plumber's 
trap used underneath sinks, creating a liquid seal to prevent noxious gases from entering a home 
from the sewer system. A loop of tube or pipe called a pigtail siphon achieves the same purpose: 

Fill valve 


Pressure gauge 


"Pigtail" siphon 

Pressure gauge 


Isolation ("block") valve 

Fill valve 

Steam pipe 



12.6.9 Mounting brackets 

An accessory specifically designed for differential pressure transmitters, but useful for other field- 
mounted instruments as well, is the 2 inch pipe mounting bracket. Such a bracket is manufactured 
from heavy-gauge sheet metal and equipped with a U-bolt designed to clamp around any 2 inch 
black iron pipe. Holes stamped in the bracket match mounting bolts on the capsule flanges of most 
common differential pressure transmitters, providing a mechanically stable means of attaching a 
differential pressure transmitter to a framework in a process area. 

The following photographs show several different instruments mounted to pipe sections using 
these brackets: 


12.7 Process/instrument suitability 

On a fundamental level, pressure is universal. Regardless of the fluid in question; liquid or gas, hot 
or cold, corrosive or inert, pressure is nothing more than the amount of force exerted by that fluid 
over a unit area: 


It should come as no surprise, then, that the common mechanical sensing elements for measuring 
pressure (bellows, diaphragm, bourdon tube, etc.) are equally applicable to all fluid pressure 
measurement applications, at least in principle. It is normally a matter of proper material selection 
and element strength (material thickness) to make a pressure instrument suitable for any range of 
process fluids. 

Fill fluids used in pressure instruments - whether it be the dielectric liquid inside a differential 
capacitance sensor, the fill liquid of a remote or chemical seal system, or liquid used to fill a vertical 
section of impulse tubing - must be chosen so as to not adversely react with or contaminate the 

Pure oxygen processes require that no system component have traces of hydrocarbon fluids 
present. While oxygen itself is not explosive, it greatly accelerates the combustion and explosive 
potential of any flammable substance. Therefore, a pressure gauge calibrated using oil as the working 
fluid in a deadweight tester would definitely not be suitable for pure oxygen service! The same may 
be said for a differential pressure transmitter with a hydrocarbon-based fill inside its pressure-sensing 
capsule 14 . 

Pharmaceutical, medical, and food manufacturing processes require strict purity and the ability 
to disinfect all elements in the process system at will. Stagnant lines are not allowed in such 
processes, as microbe cultures may flourish in such "dead end" piping. Remote seals are very 
helpful in overcoming this problem, but the fill fluids used in remote systems must be chosen so that 
a leak in the isolating diaphragm will not contaminate the process. 

Manometers, of course, are rather limited in their application, as their operation depends 
on direct contact between process fluid and manometer liquid. In the early days of industrial 
instrumentation, liquid mercury was a very common medium for process manometers, and it was 
not unusual to see a mercury manometer used in direct contact with a process fluid such as oil or 
water to provide pressure indication: 

14 Although this fluid would not normally contact pure oxygen in the process, it could if the isolating diaphragm 
inside the transmitter were to ever leak. 




Water pipe 


Isolation ("block") valve 



Range tube 

Thankfully, those days are gone. Mercury (chemical symbol "Hg") is a toxic metal and therefore 
hazardous to work with. Calibration of these manometers was also challenging due to the column 
height of the process liquid in the impulse line and the range tube. When the process fluid is a gas, 
the difference in mercury column height directly translates to sensed pressure by the hydrostatic 
pressure formula P = pgh or P = jh. When the process fluid is a liquid, though, the shifting of 
mercury columns also creates a change in height of the process liquid column, which means the 
indicated pressure is a function of the height difference (h) and the difference in density between 
the process liquid and mercury. Consequently, the indications provided by mercury manometers in 
liquid pressure applications were subject to correction according to process liquid density. 



Beckerath, Alexander von; Eberlein, Anselm; Julien, Hermann; Kersten, Peter; and Kreutzer, 
Jochem, WIKA-Handbook, Pressure and Temperature Measurement, WIKA Alexander Wiegand 
GmbH & Co., Klingenberg, Germany, 1995. 

"Digital Sensor Technology" (PowerPoint slideshow presentation), Yokogawa Corporation of 

Fribance, Austin E., Industrial Instrumentation Fundamentals, McGraw-Hill Book Company, New 
York, NY, 1962. 

Kallen, Howard P., Handbook of Instrumentation and Controls, McGraw-Hill Book Company, Inc., 
New York, NY, 1961. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Patrick, Dale R. and Patrick, Steven R., Pneumatic Instrumentation, Delmar Publishers, Inc., 
Albany, NY, 1993. 

Technical Note: "Rosemount 1199 Fill Fluid Specifications", Rosemount, Emerson Process 
Management, 2005. 

Chapter 13 

Continuous level measurement 

Many industrial processes require the accurate measurement of fluid or solid (powder, granule, 
etc.) height within a vessel. Some process vessels hold a stratified combination of fluids, naturally 
separated into different layers by virtue of differing densities, where the height of the interface point 
between liquid layers is of interest. 

A wide variety of technologies exist to measure the level of substances in a vessel, each exploiting 
a different principle of physics. This chapter explores the major level-measurement technologies in 
current use. 




13.1 Level gauges (sightglasses) 

The level gauge, or sightglass is to liquid level measurement as manometers are to pressure 
measurement: a very simple and effective technology for direct visual indication of process level. 
In its simplest form, a level gauge is nothing more than a clear tube through which process liquid 
may be seen. The following photograph shows a simple example of a sightglass: 

A functional diagram of a sightglass shows how it visually represents the level of liquid inside a 
vessel such as a storage tank: 



gauge valve 


Level gauge 

(glass tube) 

Process liquid 

Level gauge valves exist to allow replacement of the glass tube without emptying or depressurizing 
the process vessel. These valves are usually equipped with flow-limiting devices in the event of a 
tube rupture, so that too much process fluid does not escape even when the valves are fully open. 

Some level gauges called reflex gauges are equipped with special optics to facilitate the viewing 
of clear liquids, which is problematic for simple glass-tube sight glasses. 

As simple and apparently trouble-free as level gauges may seem, there are special circumstances 
where they will register incorrectly. One such circumstance is in the presence of a lighter liquid layer 
existing between the connection ports of the gauge. If a lighter (less dense) liquid exists above a 
heavier (denser) liquid in the process vessel, the level gauge may not show the proper interface, if 
at all: 

Water (only) 



Here we see how a column of water in the sightglass shows less (total) level than the combination 
of water and oil inside the process vessel. Since the oil lies between the two level gauge ports into 
the vessel (sometimes called nozzles), it cannot enter the sightglass tube, and therefore the level 
gauge will continue to show just water. 

If by chance some oil does find its way into the sightglass tube - either by the interface level 
dropping below the lower nozzle or the total level rising above the upper nozzle - the oil/water 
interface shown inside the level gauge may not continue to reflect the true interface inside the vessel 
once the interface and total levels return to their previous positions: 


In effect, the level gauge and vessel together form a U-tube manometer. So long as the pressures 
from each liquid column are the same, the columns balance each other. The problem is, many 
different liquid-liquid interface columns can have the same hydrostatic pressure without being 
identical to one another: 

-« — Oil 
-*< — Water 

-« — Oil 

-« — Water 

-« — Oil 


The only way to ensure proper two-part liquid interface level indication in a sightglass is to keep 
both ports (nozzles) submerged: 



Nozzle is submerged 

-" — Oil 



Nozzle is submerged 

Another troublesome scenario for level gauges is when the liquid inside the vessel is substantially 
hotter than the liquid in the gauge, causing the densities to be different. This is commonly seen 
on boiler level gauges, where the water inside the sightglass cools off substantially from its former 
temperature inside the boiler drum: 


Looking at the sightglass as a U-tube manometer again, we see that unequal-height liquid columns 
may indeed balance each other's hydrostatic pressures if the two columns are comprised of liquids 
with different densities. The weight density of water is 62.4 lb/ft 3 at standard temperature, but 
may be as low as only 36 lb/ft 3 at temperatures common for power generation boilers. 


13.2 Float 


Perhaps the simplest form of solid or liquid level measurement is with a float: a device that rides on 
the surface of the fluid or solid within the storage vessel. The float itself must be of substantially 
lesser density than the substance of interest, and it must not corrode or otherwise react with the 

Floats may be used for manual "gauging" of level, as illustrated here: 


A person lowers a float down into a storage vessel using a flexible measuring tape, until the tape 
goes slack due to the float coming to rest on the material surface. At that point, the person notes 
the length indicated on the tape (reading off the lip of the vessel access hole). 

Obviously, this method of level measurement is tedious and may pose risk to the person 
conducting the measurement. If the vessel is pressurized, this method is simply not applicable. 

If we automate the person's function using a small winch controlled by a computer - having 
the computer automatically lower the float down to the material surface and measure the amount 
of cable played out at each measurement cycle - we may achieve better results without human 
intervention. Such a level gauge may be enclosed in such a way to allow pressurization of the vessel, 

13.2. FLOAT 



-< — Cable 

A simpler version of this technique uses a spring-reel to constantly tension the cable holding the 
float, so that the float continuously rides on the surface of the liquid in the vessel 1 : 

cable reel 



cable reel 

The following photograph shows the "measurement head" of a spring-reel tape-and-float liquid 
level transmitter, with the vertical pipe housing the tape on its way to the top of the storage tank 
where it will turn 180 degrees via two pulleys and attach to the float inside the tank: 

1 A spring-loaded cable float only works with liquid level measurement, while a retracting float will measure liquids 
and solids with equal ease. The reason for this limitation is simple: a float that always contacts the material surface 
is likely to become buried if the material in question is a solid (powder or granules) , which must be fed into the vessel 
from above. 



The spring reel's angular position may be measured by a multi-turn potentiometer or a rotary 
encoder (located inside the "head" unit), then converted to an electronic signal for transmission to a 
remote display, control, and/or recording system. Such systems are used extensively for measurement 
of water and fuel in storage tanks. 

If the liquid inside the vessel is subject to turbulence, 
float cable in a vertical orientation: 

e wires may be necessary to keep the 

13.2. FLOAT 





Guide wire 

The guide wires are anchored to the floor and roof of the vessel, passing through ring lugs on 
the float to keep it from straying laterally. 

One of the potential disadvantages of tape-and-float level measurement systems is fouling of the 
tape (and guide wires) if the substance is sticky or unclean. 

A variation on the theme of float level measurement is to place a small float inside the tube of a 
sightglass-style level gauge: 


Level gauge 

The float's position inside the tube may be readily detected by ultrasonic waves, magnetic sensors 
or any other applicable means. Locating the float inside a tube eliminates the need for guide wires 
or a sophisticated tape retraction or tensioning system. If no visual indication is necessary, the 
level gauge tube may be constructed out of metal instead of glass, greatly reducing the risk of 


tube breakage. All the problems inherent to sightglasses, however, still apply to this form of float 


13.3 Hydrostatic pressure 

A vertical column of fluid exerts a pressure due to the column's weight. The relationship between 
column height and fluid pressure at the bottom of the column is constant for any particular fluid 
(density) regardless of vessel width or shape. 

This principle makes it possible to infer the height of liquid in a vessel by measuring the pressure 
generated at the bottom: 

Same pressure! 

The mathematical relationship between liquid column height and pressure is as follows: 

P = pgh P = -fh 


P = Hydrostatic pressure 

p = Mass density of fluid in kilograms per cubic meter (metric) or slugs per cubic foot (British) 

g = Acceleration of gravity 

7 = Weight density of fluid in newtons per cubic meter (metric) or pounds per cubic foot (British) 

h = Height of vertical fluid column above point of pressure measurement 

For example, the pressure generated by a column of oil 12 feet high having a weight density (7) 
of 40 pounds per cubic foot is: 

P = jh 
, 12 ft\ (4Q lb 

1 J \ ft 3 
480 lb 



Note the cancellation of units, resulting in a pressure value of 480 pounds per square foot (PSF). 
To convert into the more common pressure unit of pounds per square inch, we may multiply by the 
proportion of square feet to square inches, eliminating the unit of square feet by cancellation and 
leaving square inches in the denominator: 

480 lb 

l 2 it z 
12 2 in 

480 lb W 1 ft 2 




3.33 lb 


3.33 PSI 

Thus, a pressure gauge attached to the bottom of the vessel holding a 12 foot column of this oil 
would register 3.33 PSI. It is possible to customize the scale on the gauge to read directly in feet of 
oil (height) instead of PSI, for convenience of the operator who must periodically read the gauge. 
Since the mathematical relationship between oil height and pressure is both linear and direct, the 
gauge's indication will always be proportional to height. 

Any type of pressure-sensing instrument may be used as a liquid level transmitter by means of 
this principle. In the following photograph, you see a Rosemount model 1151 pressure transmitter 
being used to measure the height of colored water inside a clear plastic tube: 

The critically important factor in liquid level measurement using hydrostatic pressure is liquid 
density. One must accurately know the liquid's density in order to have any hope of measuring 
that liquid's level using hydrostatic pressure, since density is an integral part of the height/pressure 
relationship (P = pgh and P = 7/1). Having an accurate assessment of liquid density also implies that 
density must remain relatively constant despite other changes in the process. If the liquid density 
is subject to random variation, the accuracy of any hydrostatic pressure-based level instrument will 
correspondingly vary. 

It should be noted, though, that changes in liquid density will have absolutely no effect on 
hydrostatic measurement of liquid mass, so long as the vessel has a constant cross-sectional area 
throughout its entire height. A simple thought experiment proves this: imagine a vessel partially 
full of liquid, with a pressure transmitter attached to the bottom to measure hydrostatic pressure. 



Now imagine the temperature of that liquid increasing, so that its volume expands and has a lower 
density than before. Assuming no addition or loss of liquid to or from the vessel, any increase in 
liquid level will be strictly due to volume expansion (density decrease). Liquid level inside this vessel 
will rise, but the transmitter will sense the exact same hydrostatic pressure as before, since the rise 
in level is precisely countered by the decrease in density (if h increases by the same factor that 7 
decreases, then P = jh must remain the same!). In other words, hydrostatic pressure is seen to 
be a direct indication of the liquid mass contained within the vessel, regardless of changes in liquid 

Differential pressure transmitters are the most common pressure-sensing device used in this 
capacity to infer liquid level within a vessel. In the hypothetical case of the oil vessel just considered, 
the transmitter would connect to the vessel in this manner (with the high side toward the process 
and the low side vented to atmosphere): 

output signal 


Impulse tube 


Connected as such, the differential pressure transmitter functions as a gauge pressure transmitter, 
responding to hydrostatic pressure exceeding ambient (atmospheric) pressure. As liquid level 
increases, the hydrostatic pressure applied to the "high" side of the differential pressure transmitter 
also increases, driving the transmitter's output signal higher. 

Some pressure-sensing instruments are built specifically for hydrostatic measurement of liquid 
level in vessels, doing away with impulse tubing altogether in favor of a special kind of sealing 
diaphragm that protrudes slightly into the vessel through a flanged pipe entry (commonly called a 
nozzle). A photograph of such a level transmitter is shown here: 



The calibration table for a transmitter close-coupled to the bottom of an oil storage tank would 
be as follows, assuming a zero to twelve foot measurement range for oil height, an oil density of 40 
pounds per cubic foot, and a 4-20 mA transmitter output signal range: 

Oil level 

Percent of range 

Hydrostatic pressure 

Transmitter output 




4 mA 

3 ft 


0.833 PSI 

8 mA 

6 ft 


1.67 PSI 

12 mA 

9 ft 


2.50 PSI 

16 mA 

12 ft 


3.33 PSI 

20 mA 



13.3.1 Bubbler systems 

An interesting variation on this theme of direct hydrostatic pressure measurement is the use of a 
purge gas to measure hydrostatic pressure in a liquid-containing vessel. This eliminates the need for 
direct contact of the process liquid against the pressure-sensing element, which can be advantageous 
if the process liquid is corrosive. 

Such systems are often called bubble tube or dip tube systems, the former name being 
appropriately descriptive for the way purge gas bubbles out the end of the tube as it is submerged 
in process liquid. A key detail of a bubble tube system is to provide a means of limiting gas flow 
through the tube, so that the purge gas backpressure properly reflects hydrostatic pressure at the 
end of the tube with no additional pressure due to frictional losses along the length of the tube. 
Most bubble tube systems, therefore, are provided with some means of monitoring purge gas flow, 
typically with a rotameter or with a sightfeed bubbler: 

Purge supply 

Purge supply 

If the purge gas flow is not too great, gas pressure measured anywhere in the tube system 
downstream of the needle valve will be equal to the hydrostatic pressure of the process liquid at 
the bottom of the tube where the gas escapes. In other words, the purge gas acts to transmit the 
liquid's hydrostatic pressure to some remote point where a pressure-sensing instrument is located. 
A general rule-of-thumb is to limit purge gas flow to the point where you can easily count individual 
bubbles exiting the bubble tube (or inside the sightfeed bubbler if one is provided on the system) . 

As with all purged systems, certain criteria must be met for successful operation. Listed here 
are a few pertinent questions to consider for a bubble tube system: 

• How reliable is the supply of purge fluid? If this stops for any reason, the level measurement 
may be in error! 


• Is the purge fluid supply pressure guaranteed to exceed the hydrostatic pressure at all times, 
to ensure continuous purging (bubbling)? 

• What options exist for purge gases that will not adversely react with the process? 

• What options exist for purge gases that will not contaminate the process? 

• How expensive will it be to maintain this constant flow of purge gas into the process? 

One measurement artifact of a bubble tube system is a slight variation in pressure each time a 
new bubble breaks away from the end of the tube. The amount of pressure variation is approximately 
equal to the hydrostatic pressure of process fluid at a height equal to the diameter of the bubble, 
which in turn will be approximately equal to the diameter of the bubble tube. For example, a 
1/4 inch diameter dip tube will experience pressure oscillations with a peak-to-peak amplitude of 
approximately 1/4 inch elevation of process liquid. The frequency of this pressure oscillation, of 
course, will be equal to the rate at which individual bubbles escape out the end of the dip tube. 

Usually, this is a small variation when considered in the context of the measured liquid height in 
the vessel. A pressure oscillation of approximately 1/4 inch compared to a measurement range of 
to 10 feet, for example, is only about 0.2% of span. Modern pressure transmitters have the ability 
to "filter" or "dampen" pressure variations over time, which is a useful feature for minimizing the 
effect such a pressure variation will have on system performance. 



13.3.2 Transmitter suppression and elevation 

A very common scenario for liquid level measurement is where the pressure-sensing instrument is 
not located at the same level as the 0% measurement point. The following photograph shows an 
example of this, where a Rosemount model 3051 differential pressure transmitter is being used to 
sense hydrostatic pressure of colored water inside a (clear) vertical plastic tube: 

Consider the example of a pressure sensor measuring the level of liquid ethanol in a storage 
tank. The measurement range for liquid height in this ethanol storage tank is to 40 feet, but the 
transmitter is located 30 feet below the tank: 



(vent) I 

100% -r 

span = 40 ft 


y= 49.3 lb/ft 3 


30 ft 

This means the transmitter's impulse line contains a 30-foot elevation head of ethanol, so that 
the transmitter "sees" 30 feet of ethanol when the tank is empty and 70 feet of ethanol when the 
tank is full. A 3-point calibration table for this instrument would look like this, assuming a 4 to 20 
mA DC output signal range: 

Ethanol level 
in tank 

Percent of 

(inches of water) 







284 "W.C. 

10.3 PSI 

4 mA 

20 ft 


474 "W.C. 

17.1 PSI 

12 mA 

40 ft 


663 "W.C. 

24.0 PSI 

20 mA 

Another common scenario is where the transmitter is mounted at or near the vessel's bottom, 
but the desired level measurement range does not extend to the vessel bottom: 




span = 5 ft 


4 ft 

(vent) 1 

Castor oil 
y= 60.5 lb/ft 3 


In this example, the transmitter is mounted exactly at the same level as the vessel bottom, but 
the level measurement range goes from 4 feet to 9 feet (a 5 foot span). At the level of castor oil 
deemed 0%, the transmitter "sees" a hydrostatic pressure of 1.68 PSI (46.5 inches of water column) 
and at the 100% castor oil level the transmitter "sees" a pressure of 3.78 PSI (105 inches water 
column). Thus, these two pressure values would define the transmitter's lower and upper range 
values (LRV and URV), respectively. 

The term for describing either of the previous scenarios, where the lower range value (LRV) of 
the transmitter's calibration is a positive number, is called zero suppression 2 . If the zero offset is 
reversed (e.g. the transmitter mounted at a location higher than the 0% process level), it is referred 
to as zero elevation 3 . 

If the transmitter is elevated above the process connection point, it will most likely "see" a 
negative pressure (vacuum) with an empty vessel owing to the pull of liquid in the line leading down 
from the instrument to the vessel. It is vitally important in elevated transmitter installations to use 
a remote seal rather than an open impulse line, so that liquid cannot dribble out of this line and 
into the vessel 4 : 

2 Or alternatively, zero depression. 

3 There is some disagreement among instrumentation professionals as to the definitions of these two terms. 
According to Bela G. Liptak's Instrument Engineers' Handbook, Process Measurement and Analysis (Fourth Edition, 
page 67), "suppressed zero range" refers to the transmitter being located below the 0% level (the LRV being a positive 
pressure value), while "suppression," "suppressed range," and "suppressed span" mean exactly the opposite (LRV is a 
negative value). The Yokogawa Corporation defines "suppression" as a condition where the LRV is a positive pressure 
("Autolevel" Application Note), as does the Michael MacBeth in his CANDU Instrumentation &: Control course (lesson 
1, module 4, page 12), Foxboro's technical notes on bubble tube installations (pages 4 through 7), and Rosemount's 
product manual for their 1151 Alphaline pressure transmitter (page 3-7). Interestingly, the Rosemount document 
defines "zero range suppression" as synonymous with "suppression," which disagrees with Liptak's distinction. My 
advice: draw a picture if you want the other person to clearly understand what you mean! 

4 As you are about to see, the calibration of an elevated transmitter depends on us knowing how much hydrostatic 
pressure (or vacuum, in this case) is generated within the tube connecting the transmitter to the process vessel. If 
liquid were to ever escape from this tube, the hydrostatic pressure would be unpredictable, and so would be the 
accuracy of our transmitter as a level-measuring instrument. A remote seal diaphragm guarantees no fill fluid will be 
lost if and when the process vessel goes empty. 



100% -r 

span = 1 1 ft 



Sea water 
y= 64.0 lb/ft 3 


6 ft 

■ Capillary tube with 
fill fluid y= 58.3 lb/ft 3 


Remote seal 

In this example, we see a remote seal system with a fill fluid having a density of 58.3 lb/ft 3 , and 
a process level measurement range of to 11 feet of sea water (density = 64 lb/ft 3 ). The transmitter 
elevation is 6 feet, which means it will "see" a vacuum of -2.43 PSI (-67.2 inches of water column) 
when the vessel is completely empty. This, of course, will be the transmitter's calibrated lower range 
value (LRV). The upper range value (URV) will be the pressure "seen" with 11 feet of sea water in 
the vessel. This much sea water will contribute an additional 4.89 PSI of hydrostatic pressure at the 
level of the remote seal diaphragm, causing the transmitter to experience a pressure of +2.46 PSI 5 . 

5 The sea water's positive pressure at the remote seal diaphragm adds to the negative pressure already generated 
by the downward length of the capillary tube's fill fluid (-2.43 PSI), which explains why the transmitter only "sees" 
2.46 PSI of pressure at the 100% full mark. 



13.3.3 Compensated leg systems 

The simple and direct relationship between liquid height in a vessel and pressure at the bottom 
of that vessel is ruined if another source of pressure exists inside the vessel other than hydrostatic 
(elevation head). This is virtually guaranteed to be the case if the vessel in question is unvented. 
Any gas or vapor pressure accumulation in an enclosed vessel will add to the hydrostatic pressure 
at the bottom, causing any pressure-sensing instrument to falsely register a high level: 

output signal 


Pressure = P + yli 

A pressure transmitter has no way of "knowing" how much of the sensed pressure is due to liquid 
elevation and how much of it is due to pressure existing in the vapor space above the liquid. Unless 
a way can be found to compensate for any non-hydrostatic pressure in the vessel, this extra pressure 
will be interpreted by the transmitter as additional liquid level. 

Moreover, this error will change as gas pressure inside the vessel changes, so it cannot simply 
be "calibrated out" by a static zero shift within the instrument. The only way to hydrostatically 
measure liquid level inside an enclosed (non-vented) vessel is to continuously compensate for gas 

Fortunately, the capabilities of a differential pressure transmitter make this a simple task. All 
we need to do is connect a second impulse line (called a compensating leg), from the "Low" port 
of the transmitter to the top of the vessel, so that the "Low" side of the transmitter experiences 
nothing but the gas pressure enclosed by the vessel, while the "High" side experiences the sum of 
gas and hydrostatic pressures. Since a differential pressure transmitter responds only to differences 
in pressure between "High" and "Low" sides, it will naturally subtract the gas pressure (P ga s) to 
yield a measurement based solely on hydrostatic pressure (7/1): 



Gas pressure 

(P g as) 

Density = y 



/ le 9 

output signal 

Pressure = P + yh Pressure = P 


(Pgas + lh) ~ P, 



The amount of gas pressure inside the vessel now becomes completely irrelevant to the 
transmitter's indication, because its effect is canceled at the differential pressure instrument's sensing 
element. If gas pressure inside the vessel were to increase while liquid level remained constant, the 
pressure sensed at both ports of the differential pressure transmitter would increase by the exact 
same amount, with the pressure difference between the "high" and "low" ports remaining absolutely 
constant with the constant liquid level. This means the instrument's output signal is a representation 
of hydrostatic pressure only, which represents liquid height (assuming a known liquid density 7). 

Unfortunately, it is common for enclosed vessels to hold condensible vapors, which may over time 
fill a compensating leg full of liquid. If the tube connecting the "Low" side of a differential pressure 
transmitter fills completely with a liquid, this will add a hydrostatic pressure to that side of the 
transmitter, causing another calibration shift. This wet leg condition makes level measurement more 
complicated than a dry leg condition where the only pressure sensed by the transmitter's "Low" side 
is gas pressure (P gas )- 




Gas pressure 

I gas) 

Density = y, 

leg (wet) 

Density = y. 

output signal 

Pressure = P + y 1 h 1 Pressure = P + y 2 h 

■ l\hl) - (Pgas + 72^2) = 71^1 - 72^-2 

Gas pressure still cancels due to the differential nature of the pressure transmitter, but now the 
transmitter's output indicates a difference of hydrostatic pressures between the vessel and the wet 
leg, rather than just the hydrostatic pressure of the vessel's liquid level. Fortunately, the hydrostatic 
pressure generated by the wet leg will be constant, so long as the density of the condensed vapors 
filling that leg (72) is constant. If the wet leg's hydrostatic pressure is constant, we can compensate 
for it by calibrating the transmitter with an intentional zero shift, so that it indicates as though it 
were measuring hydrostatic pressure on a vented vessel. 

Differential pressure =71/11 — Constant 

We may ensure a constant density of wet leg liquid by intentionally filling that leg with a liquid 
known to be denser than the densest condensed vapor inside the vessel. We could also use a 
differential pressure transmitter with remote seals and capillary tubes filled with liquid of known 



Fill valve 

Gas pressure 

(with condensible vapors) 


Gas pressure 

(with condensible vapors) 

Capillary tube 

The following example shows the calibration table for a compensated-leg (wet) hydrostatic level 
measurement system, for a gasoline storage vessel and water as the wet leg fill fluid. Here, I 
am assuming a density of 41.0 lb/ft 3 for gasoline and 62.4 lb/ft 3 for water, with a to 10 foot 
measurement range and an 11 foot wet leg height: 

Gasoline level 

Percent of range 

Pressure at transmitter 

Transmitter output 



-4.77 PSI 

4 niA 

2.5 ft 


-4.05 PSI 

8 niA 

5 ft 


-3.34 PSI 

12 mA 

7.5 ft 


-2.63 PSI 

16 mA 

10 ft 


-1.92 PSI 

20 mA 

Note that due to the superior density and height of the wet (water) leg, the transmitter always 
sees a negative pressure (pressure on the "Low" side exceeds pressure on the "High" side). With 
some older differential pressure transmitter designs, this was a problem. Consequently, it is common 
to see "wet leg" hydrostatic transmitters installed with the "Low" port connected to the bottom of 
the vessel and the "High" port connected to the compensating leg. In fact, it is still common to see 
modern differential pressure transmitters installed in this manner 6 , although modern transmitters 
may be calibrated for negative pressures just as easily as for positive pressures. It is vitally important 
to recognize that any differential pressure transmitter connected as such (for any reason) will respond 
in reverse fashion to increases in liquid level. That is to say, as the liquid level in the vessel rises, 
the transmitter's output signal will decrease instead of increase: 

6 Sometimes this is done out of habit, other times because instrument technicians do not know the capabilities of 
new technology. 



High side of DP transmitter connected 
to the compensating impulse leg 

Signal decreases with 
increasing liquid level! 

Either way of connecting the transmitter to the vessel will suffice for measuring liquid level, so 
long as the instrumentation receiving the transmitter's signal is properly configured to interpret the 
signal. The choice of which way to connect the transmitter to the vessel should be driven by fail-safe 
system design, which means to design the measurement system such that the most probably system 
failures - including broken signal wires - result in the control system "seeing" the most dangerous 
process condition and therefore taking the safest action. 



13.3.4 Tank expert systems 

An alternative to using a compensating leg to subtract gas pressure inside an enclosed vessel is to 
simply use a second pressure transmitter and electronically subtract the two pressures in a computing 

Gas pressure 

( * gas) 




Pressure = P„ 




LY >v.i->Height 



Pressure = P + yh 

This approach enjoys the distinct advantage of avoiding a potentially wet compensating leg, but 
suffers the disadvantages of extra cost and greater error due to the potential calibration drift of two 
transmitters rather than just one. Such a system is also impractical in applications where the gas 
pressure is substantial compared to the hydrostatic (elevation head) pressure 7 . 

If we add a third pressure transmitter to this system, located a known distance (x) above the 
bottom transmitter, we have all the pieces necessary for what is called a tank expert system. These 
systems are used on large storage tanks operating at or near atmospheric pressure, and have the 
ability to measure infer liquid height, liquid density, total liquid volume, and total liquid mass stored 
in the tank: 

7 This is due to limited transmitter resolution. Imagine an application where the elevation head was 10 PSI 
(maximum) yet the vapor space pressure was 200 PSI. The majority of each transmitter's working range would be 
"consumed" measuring gas pressure, with hydrostatic head being a mere 5% of the measurement range. This would 
make precise measurement of liquid level very difficult, akin to trying to measure the sound intensity of a whisper in 
a noisy room. 



A "tank expert" system 

Gas pressure 



Density = y 




Pressure = P„ 




LY ) > Height 




Pressure = P + y(h - x) 


Pressure = P + yh 

The pressure difference between the bottom and middle transmitters will change only if the liquid 
density changes 8 , since the two transmitters are separated by a known and fixed height difference. 
This allows the level computer (LY) to continuously calculate liquid density (7): 

{P gas + 7/1) - [P gas + j(h - x)] 



Pgas +jh~ Pgas ~ l{h ~ x) 


Pgas +jh- Pgas ~ 7^ + jx) 






8 Assuming the liquid level is equal to or greater than x. Otherwise, the pressure difference between Pbottom an d 
Pmiddle will depend on liquid density and liquid height. However, this condition is easy to check: the level computer 
simply checks to see if P m iddle and Ptop are unequal. If so, then the computer knows the liquid level exceeds x and it 
is safe to calculate density. If not, and P m iddle registers the same as Pt op , the computer knows those two transmitters 
are both registering gas pressure only, and it knows to stop calculating density. 


Once the computer knows the value of 7, it may calculate the height of liquid in the tank with 
great accuracy based on the pressure measurements taken by the bottom and top transmitters: 

* bottom *top \*gas ' / " ) gas 

Pbottom — Ptop = "fh 
1 bottom ^top 7 


With all the computing power available in the LY, it is possible to characterize the tank such 
that this height measurement converts to a precise volume measurement 9 (V), which may then be 
converted into a total mass (to) measurement based on the mass density of the liquid (p) and the 
acceleration of gravity (g). First, the computer calculates mass density based on the proportionality 
between mass and weight (shown here starting with the equivalence between the two forms of the 
hydrostatic pressure formula): 

pgh = jh 

99 = 1 


9 = ~ 


Armed with the mass density of the liquid inside the tank, the computer may now calculate total 
liquid mass stored inside the tank: 

to = pV 

Dimensional analysis shows how units of mass density and volume cancel to yield only units of 
mass in this last equation: 




Here we see a vivid example of how several measurements may be inferred from just a few actual 
process (in this case, pressure) measurements. Three pressure measurements on this tank allow us 
to compute four inferred variables: liquid density, liquid height, liquid volume, and liquid mass. 

The accurate measurement of liquids in storage tanks is not just useful for process operations, 
but also for conducting business affairs. Whether the liquid represents raw material purchased from 
a supplier, or a processed product ready to be pumped out to a customer, both parties have a vested 
interest in knowing the exact quantity of liquid bought or sold. Measurement applications such as 

9 The details of this math depend entirely on the shape of the tank. For vertical cylinders — the most common 
shape for vented storage tanks - volume and height are related by the simple formula V = Txr 2 h where r is the radius 
of the tank's circular base. Other tank shapes and orientations may require much more sophisticated formulae to 
calculate stored volume from height. See section 17.2, beginning on page 583, for more details on this subject. 


this are known as custody transfer, because they represent the transfer of custody (ownership) of a 
substance exchanged in a business agreement. It is common for both buyer and seller to operate and 
maintain their own custody transfer instrumentation, and to compare the instruments' readings for 
concurrence within a mutually agreed margin of error. 



13.3.5 Hydrostatic interface level measurement 

Hydrostatic pressure sensors may be used to detect the level of a liquid-liquid interface, if and 
only if the total height of liquid sensed by the instrument is fixed. A single hydrostatic-based level 
instrument cannot discern between a changing interface level and a changing total level, so the latter 
must be fixed in order to measure the former. 

One way of fixing total liquid height is to equip the vessel with an overflow pipe, and ensure 
that drain flow is always less than incoming flow (so that some flow must always go through the 
overflow pipe). This strategy naturally lends itself to separation processes, where a mixture of light 
and heavy liquids are separated by their differing densities: 

Inlet pipe 


h 2 


Light liquid 

Density = y 2 

Heavy liquid 

Density = y } 


""^ pipe 
(light liquid out) 



output signal 



Pressure = y i h 1 + y 2 h 2 

Drain pipe 
(heavy liquid out) 

Here we see a practical application for liquid- liquid interface level measurement. If the goal is to 
separate two liquids of differing densities from one another, we need only the light liquid to exit out 
the overflow pipe and only the heavy liquid to exit out the drain pipe. This means we must control 
the interface level to stay between those two piping points on the vessel. If the interface drifts too 
far up, heavy liquid will be carried out the overflow pipe; and if we let the interface drift too far 
down, light liquid will flow out of the drain pipe. The first step in controlling any process variable 
is to measure that variable, and so here we are faced with the necessity of measuring the interface 
point between the light and heavy liquids. 

Another way of fixing the total height seen by the transmitter is to use a compensating leg located 
at a point on the vessel always lower than the total liquid height. In this example, a transmitter 
with remote seals is used: 



Inlet pipe 


Light liquid 

Density = y 2 

Heavy liquid 

Density = y } 

-> (light liquid out) 

Fill fluid 
Density =y 4 








output signal 

Drain pipe 
(heavy liquid out) 


Pressure = y y /z 7 + y 2 h 2 + y 2 h 3 

Pressure = y 4 h 4 + y 2 h 3 

Since both sides of the differential pressure transmitter "see" the hydrostatic pressure generated 
by the liquid column above the top connection point (72/13), this term is naturally canceled: 

(71/11 + 72/J2 + 72^3) - (74^4 + 72/13) 

7i hi + 72^2 + 72/13 - 74/14 - 72 h 3 

Jihi +72/12 - 74/14 

The hydrostatic pressure in the compensating leg is constant (74/14 = Constant), since the fill 
fluid never changes density and the height never changes. This means the transmitter's sensed 
pressure will differ from that of an uncompensated transmitter merely by a constant offset, which 
may be "calibrated out" so as to have no impact on the measurement: 

7i hi + 72/12 — Constant 

At first, it may seem as though determining the calibration points (lower- and upper-range 
values: LRV and URV) for a hydrostatic interface level transmitter is impossibly daunting given all 
the different pressures involved. A recommended problem-solving technique to apply here is that 
of a thought experiment, where we imagine what the process will "look like" at lower-range value 
condition and at the upper-range value condition, drawing two separate illustrations. 

For example, suppose we must calibrate a differential pressure transmitter to measure the 
interface level between two liquids having specific gravities of 1.1 and 0.78, respectively, over a 



span of 3 feet. The transmitter is equipped with remote seals, each containing a halocarbon fill fluid 
with a specific gravity of 1.09. The physical layout of the system is as follows: 

Light liquid 

S.G. = 0.78 

1.5 ft 


Fill fluid 
S.G. = 1.09 


output signal 

As the first step in our "thought experiment," we imagine what the process would look like with 
the interface at the LRV level, calculating hydrostatic pressures seen at each side of the transmitter: 

Interface level = LRV 

Light liquid 

S.G. = 0.78 

1.5 ft 


Fill fluid 
S.G. = 1.09 


output signal 

We know from our previous exploration of this setup that any hydrostatic pressure resulting from 
liquid level above the top remote seal location is irrelevant to the transmitter, since it is "seen" on 
both sides of the transmitter and thus cancels out. All we must do, then, is calculate hydrostatic 
pressures as though the total liquid level stopped at that upper diaphragm connection point. 

First, calculating the hydrostatic pressure "seen" at the high port of the transmitter 10 : 

10 Here I will calculate all hydrostatic pressures in units of inches water column. This is relatively easy because we 
have been given the specific gravities of each liquid, which make it easy to translate actual liquid column height into 
column heights of pure water. 



Phigh = 4.5 feet of heavy liquid + 4.5 feet of light liquid 

Phigh = 54 inches of heavy liquid + 54 inches of light liquid 

Phigh "W.C. = (54 inches of heavy liquid )(1.1) + (54 inches of light liquid )(0.78) 

Phigh "W.C. = 59.4 "W.C. +42.12 "W.C. 

P hig h = 101.52 "W.C. 
Next, calculating the hydrostatic pressure "seen" at the low port of the transmitter: 



9 feet of fill fluid 

108 inches of fill fluid 

P low "W.C. = (108 inches of fill fluid )(1.09) 


117.72 "W.C. 

The differential pressure applied to the transmitter in this condition is the difference between 
the high and low port pressures, which becomes the lower range value (LRV) for calibration: 

P LRV = 101.52 "W.C. - 117.72 "W.C. = -16.2 "W.C. 

As the second step in our "thought experiment," we imagine what the process would look like with 
the interface at the URV level, calculating hydrostatic pressures seen at each side of the transmitter: 

Interface level = URV 

Light liquid 

S.G. = 0.78 

Heavy liquid 

S.G. = 1.1 

1.5 ft 


-T- URV 

3 It 

-J- LRV 

4.5 ft 


Fill fluid 
S.G. = 1.09 




9 ft 

output signal 



Phigh = 7.5 feet of heavy liquid + 1.5 feet of light liquid 


90 inches of heavy liquid + 18 inches of light liquid 

Phigh "W.C. = (90 inches of heavy liquid )(1.1) + (18 inches of light liquid )(0.78) 

Phigh "W.C. = 99 "W.C. + 14.04 "W.C. 



113.04 "W.C. 

The hydrostatic pressure of the compensating leg is exactly the same as it was before: 9 feet of 
fill fluid having a specific gravity of 1.09, which means there is no need to calculate it again. It will 
still be 117.72 inches of water column. Thus, the differential pressure at the URV point is: 


113.04 "W.C. - 117.72 "W.C. 

-4.68 "W.C. 

Using these two pressure values and some interpolation, we may generate a 5-point calibration 
table (assuming a 4-20 mA transmitter output signal range) for this interface level measurement 

Interface level 

Percent of range 

Pressure at transmitter 

Transmitter output 

4.5 ft 


-16.2 "W.C. 

4 mA 

5.25 ft 


-13.32 "W.C. 

8 mA 

6 ft 


-10.44 "W.C. 

12 mA 

6.75 ft 


-7.56 "W.C. 

16 mA 

7.5 ft 


-4.68 "W.C. 

20 mA 

When the time comes to bench-calibrate this instrument in the shop, the easiest way to do so 
will be to set the two remote diaphragms on the workbench (at the same level), then apply 16.2 to 
4.68 inches of water column pressure to the low remote seal diaphragm with the other diaphragm 
at atmospheric pressure to simulate the desired range of negative differential pressures 11 . 

The more mathematically inclined reader will notice that the span of this instrument (URV — 
LRV) is equal to the span of the interface level (3 feet, or 36 inches) multiplied by the difference in 
specific gravities (1.1 — 0.78): 

Span in "W.C. = (36 inches) (1.1 - 0.78) 

11 Remember that a differential pressure instrument cannot "tell the difference" between a positive pressure applied 
to the low side, an equal vacuum applied to the high side, or an equivalent difference of two positive pressures with 
the low side's pressure exceeding the high side's pressure. Simulating the exact process pressures experienced in the 
field to a transmitter on a workbench would be exceedingly complicated, so we "cheat" by simplifying the calibration 
setup and applying the equivalent difference of pressure only to the "low" side. 



Span= 11.52 "W.C. 

Looking at our two "thought experiment" illustrations, we see that the only difference between 
the two scenarios is the type of liquid filling that 3-foot region between the LRV and URV marks. 
Therefore, the only difference between the transmitter's pressures in those two conditions will be 
the difference in height multiplied by the difference in density. Not only is this an easy way for us 
to quickly calculate the necessary transmitter span, but it also is a way for us to check our previous 
work: we see that the difference between the LRV and URV pressures is indeed a difference of 11.52 
inches water column just as this method predicts. 

Interface level = LRV 

Interface level = URV 

Light liquid 

S.G. = 0.78 

Heavy liquid 

S.G. = 1.1 

1.5 ft 


3 ft 

-J- LRV 

4.5 ft 


Fill fluid 
S.G. = 1.09 


9 ft 

Light liquid 

S.G. = 0.78 

Heavy liquid 

S.G. = 1.1 

1.5 ft 


3 ft 

-J- LRV 

4.5 ft 


Fill fluid 
S.G. = 1.09 


9 ft 



13.4 Displacement 

Displacer level instruments exploit Archimedes' Principle to detect liquid level by continuously 
measuring the weight of a rod immersed in the process liquid. As liquid level increases, the displacer 
rod experiences a greater buoyant force, making it appear lighter to the sensing instrument, which 
interprets the loss of weight as an increase in level and transmits a proportional output signal. 

In practice a displacer level instrument usually takes the following form: 

















The following photograph shows a Fisher "LevelTrol" model pneumatic transmitter measuring 
condensate level in a knockout drum for natural gas service. The instrument itself appears on the 
right-hand side of the photo, topped by a grey-colored "head" with two pneumatic pressure gauges 
visible. The displacer "cage" is the vertical pipe immediately behind and below the head unit. Note 
that a sightglass level gauge appears on the left-hand side of the knockout chamber (or condensate 
boot) for visual indication of condensate level inside the process vessel: 



Two photos of a disassembled LevelTrol displacer instrument appear here, showing how the 
displacer fits inside the cage pipe: 



The cage pipe is coupled to the process vessel through two block valves, allowing isolation from 
the process. A drain valve allows the cage to be emptied of process liquid for instrument service 
and zero calibration. Full-range calibration may be done by flooding the cage with process liquid (a 
wet calibration), or by suspending the displacer with a string and precise scale (a dry calibration), 
pulling upward on the displacer at just the right amount to simulate buoyancy at 100% liquid level: 




Pull up on string 
until scale registers 
the desired force 





Liquid drained 
out of cage 

valve open 

Calculation of this buoyant force is a simple matter. According to Archimedes' Principle, buoyant 
force is always equal to the weight of the fluid volume displaced. In the case of a displacer-based level 
instrument at full range, this usually means the entire volume of the displacer element is submerged 
in the liquid. Simply calculate the volume of the displacer (if it is a cylinder, V = irr 2 l, where r is 
the cylinder radius and I is the cylinder length) and multiply that volume by the weight density (7) : 




77IT / 

For example, if the weight density of the process fluid is 57.3 pounds per cubic foot and the 
displacer is a cylinder measuring 3 inches in diameter and 24 inches in length, the necessary force 
to simulate a condition of buoyancy at full level may be calculated as follows: 

57.3 lb 

1 ft" 
12 3 ir 




V = nr z l = tt(1.5 in) 2 (24 in) = 169.6 in d 

F bu0 yant = lV = fo.0332^ J (169.6 in 3 ) = 5.63 lb 

Note how important it is to maintain consistency of units! The liquid density was given in units 
of pounds per cubic foot and the displacer dimensions in inches, which would have caused serious 
problems without a conversion between feet and inches. In my example work, I opted to convert 
density into units of pounds per cubic inch, but I could have just as easily converted the displacer 
dimensions into feet to arrive at a displacer volume in units of cubic feet. 



13.4.1 Displacement interface level measurement 

Displacer level instruments may be used to measure liquid-liquid interfaces just the same as 
hydrostatic pressure instruments. One important requirement is that the displacer always be fully 
submerged. If this rule is violated, the instrument will not be able to "tell" the difference between 
a low (total) liquid level and a low interface level. 

If the displacer instrument has its own "cage," it is important that both pipes connecting the cage 
to the process vessel (sometimes called "nozzles") be submerged. This ensures the liquid interface 
inside the cage matches the interface inside the vessel. If the upper nozzle ever goes dry, the same 
problem can happen with a caged displacer instrument as with a "sightglass" level gauge (see page 
350 for a detailed explanation of this problem.). 

Calculating buoyant force on a displacer element due to a combination of two liquids is not as 
difficult as it may sound. Archimedes' Principle still holds: that buoyant force is equal to the weight 
of the fluid(s) displaced. All we need to do is calculate the combined weights and volumes of the 
displaced liquids to calculate buoyant force. For a single liquid, buoyant force is equal to the weight 
density of that liquid (7) multiplied by the volume displaced (V): 


7 y 

For a two-liquid interface, the buoyant force is equal to the sum of the two liquid weights 
displaced, each liquid weight term being equal to the weight density of that liquid multiplied by 
the displaced volume of that liquid: 


JlVy +72V2 

Assuming a displacer of constant cross-sectional area throughout its length, the volume for each 
liquid's displacement is simply equal to the same area (irr 2 ) multiplied by the length of the displacer 
submerged in that liquid: 


Heavy liquid 

Density = y 7 

IT Displacer area = nr 2 



Fbuoyant = ^/lTT^h + ^ 2 TTr 2 l 2 

Since the area (irr 2 ) is common to both buoyancy terms in this equation, we may factor it out 
for simplicity's sake: 

Fbuoyant = TTr 2 (jili + J 2 l 2 ) 

Calculating the LRV buoyant force is as simple as setting l\ equal to zero and l 2 equal to the 
displacer length (L): 

Fbuoyant (LRV) = T7r 2 "f 2 L 

Calculating the URV buoyant force is as simple as setting l 2 equal to zero and l\ equal to the 
displacer length (L): 

Fbuoyant (URV) = lTr 2 ^iL 

The buoyancy for any measurement percentage between the LRV (0%) and URV (100%) may 
be calculated by interpolation. Sample calculations are shown below for a displacer instrument 
measuring the interface level between two liquids having specific gravities of 0.850 and 1.10, with a 
displacer length of 30 inches and a displacer diameter of 2.75 inches (radius = 1.375 inches): 

/ lb \ , lb lb 

7i = 62.4 -^ (1.10) = 68.6 -^ = 0.0397 -^ 

V ft 3 / ft 3 in 3 

/ lb \ , lb lb 

72 = 62.4 -* (0.85 = 53.0 -^ = 0.0307 ^r 

V ft 3 / ft 3 in 3 

./■},„„„„„; (LRV) = -,i-(J.:-JTo iu)- j 0.0307 ^ ) (30 in) = 5.4', lb 
Fbuoyant (URV) = tt(1.375 in) 1 ' ( 0.0397 — ) (30 in) = 7.08 lb 

Interface level (inches) 

Buoyant force (pounds) 










13.5. ECHO 389 

13.5 Echo 

A completely different way of measuring liquid level in vessels is to bounce a traveling wave off the 
surface of the liquid - typically from a location at the top of the vessel - using the time-of-fllght for 
the waves as an indicator of distance, and therefore an indicator of liquid height inside the vessel. 
Echo-based level instruments enjoy the distinct advantage of immunity to changes in liquid density, 
a factor crucial to the accurate calibration of hydrostatic and displacement level instruments. In 
this regard, they are quite comparable with float-based level measurement systems. 

The single most important factor to the accuracy of an echo-based level instrument is the speed 
at which the wave travels en route to the liquid surface and back. This wave propagation speed 
is as fundamental to the accuracy of an echo instrument as liquid density is to the accuracy of 
a hydrostatic or displacer instrument. So long as this velocity is known and stable, good level 
measurement accuracy is generally easy to achieve. 

From a historical perspective, hydrostatic and displacement level instruments have a richer 
pedigree. These instruments are simpler in nature than echo-based instruments, and were 
practical long before the advent of modern electronic technology. Echo-based instruments require 
precision timing and wave-shaping circuitry, plus sensitive (and rugged!) transceiver elements, 
demanding a much higher level of technology. However, modern electronic design and instrument 
manufacturing practices are making echo-based level instruments more and more practical for 
industrial applications. At the time of this writing (2008), it is common practice in some industries 
to replace old displacer level instruments with guided-wave radar instruments, even in demanding 
applications operating at high pressures 12 . 

Liquid-liquid interfaces may also be measured with some types of echo-based level instruments, 
most commonly guided-wave radar. 

Echo-based level instruments may be "fooled" by layers of foam resting on top of the liquid, and 
the liquid-to-liquid interface detection models may have difficulty detecting non-distinct interfaces 
(such as emulsions) . Irregular structures residing within the vapor space of a vessel (such as access 
portals, mixer paddles and shafts, ladders, etc.) may wreak havoc with echo-based level instruments 
by casting false echoes back to the instrument, although this problem may be mitigated by installing 
guide tubes for the waves to travel in, or using wave probes as in the cases of guided-wave radar 
instruments. Liquid streams pouring in to the vessel through the vapor space may similarly cause 
problems for an echo instrument. Additionally, all echo-based instruments have dead zones where 
liquid level is too close to the transceiver to be accurately measured or even detected (the echo 
time-of-flight being too short for the receiving electronics to distinguish from the incident pulse). 

12 My own experience with this trend is within the oil refining industry, where legacy displacer instruments (typically 
Fisher brand "LevelTrol" units) are being replaced with new guided-wave radar transmitters, both for single-liquid 
and liquid-liquid interface applications. 



13.5.1 Ultrasonic level measurement 

Ultrasonic level instruments measure the distance from the transmitter (located at some high point) 
to the surface of a process material located further below. The time-of-flight for a sound pulse 
indicates this distance, and is interpreted by the transmitter electronics as process level. These 
transmitters may output a signal corresponding either to the fullness of the vessel (fillage) or the 
amount of empty space remaining at the top of a vessel (ullage). 







Ullage is the "natural" mode of measurement for this sort of level instrument, because the sound 
wave's time-of-flight is a direct function of how much empty space exists between the liquid surface 
and the top of the vessel. Total tank height will always be the sum of fillage and ullage, though. If 
the ultrasonic level transmitter is programmed with the vessel's total height, it may calculate fillage 
via simple subtraction: 

Fillage = Total height — Ullage 

The instrument itself consists of an electronics module containing all the power, computation, 
and signal processing circuits; plus an ultrasonic transducer to send and receive the sound waves. 
This transducer is typically piezoelectric in nature, being the equivalent of a very high-frequency 
audio speaker. A typical example is shown in the following photograph: 

13.5. ECHO 


If the ultrasonic transducer is rugged enough, and the process vessel sufficiently free of sludge 
and other sound-dampening materials accumulating at the vessel bottom, the transducer may be 
mounted at the bottom of the vessel, bouncing sound waves off the liquid surface through the liquid 
itself rather than through the vapor space: 



1 L_ 



This arrangement makes fillage the natural measurement, and ullage a derived measurement 
(calculated by subtraction from total vessel height). 

Ullage = Total height — Fillage 

Whether the ultrasonic transducer is mounted above or below the liquid level, the principle of 
detection is any significant difference in material density. If the detection interface is between a gas 
and a liquid, the abrupt change in density is enough to create a strong reflected signal. However, it 
is possible for foam and floating solids to also cause echos when the transducer is above-mounted, 
which may or may not be desirable depending on the application 13 . 

Ultrasonic level instruments enjoy the advantage of being able to measure the height of solid 
materials such as powders and grains stored in vessels, not just liquids. Certain challenges unique 
to these level measurement applications include low material density (not causing strong reflections) 
and uneven profiles (causing reflections to be scattered laterally instead of straight back to the 
ultrasonic instrument. A classic problem encountered when measuring the level of a powdered or 
granular material in a vessel is the angle of repose formed by the material as a result of being fed 
into the vessel at one point: 

13 If the goal is to only detect the liquid, then reflections from foam or solids would be bad. However, if the goal of 
measuring level is to prevent a vessel from overflowing, it is good to measure anything floating on the liquid surface! 

13.5. ECHO 


Feed chute 

Angle of repose 

This angled surface is difficult for an ultrasonic device to detect because it tends to scatter the 
sound waves laterally instead of reflecting them strongly back toward the instrument. However, even 
if the scattering problem is not significant, there still remains the problem of interpretation: what 
is the instrument actually measuring? The detected level near the vessel wall will certainly register 
less than at the center, but the level detected mid-way between the vessel wall and vessel center 
may not be an accurate average of those two heights. Moreover, this angle may decrease over time 
if mechanical vibrations cause the material to "flow" and tumble from center to edge. 

In fact, the angle will probably reverse itself if the vessel empties from a center-located chute: 

> Outlet 

For this reason, solids storage measurement applications demanding high accuracy generally use 
other techniques, such as weight-based measurement (see section 13.7 for more information). 


Since the speed of sound is so important to accurate distance calculations for ultrasonic 
instruments, some ultrasonic level instruments are equipped with temperature sensors to measure 
the temperature of the fluid through which the sound waves travel. A formula programmed into 
the transmitter calculates the speed of sound based on temperature, so that the instrument may 
continuously compensate for changes in sound velocity rooted in temperature changes, and therefore 
maintain superior accuracy over a wide range of ambient conditions. In the vast majority of 
ultrasonic level transmitter installations (where the instrument is mounted above the liquid level 
such that the sound waves travel through air, bounce off liquid, and travel back through air), it 
is the speed of sound through air that matters. The speed of sound through liquid is irrelevant 
in these applications, since most of the acoustic energy reflects off the liquid surface and therefore 
never travels through it. 

13.5. ECHO 


13.5.2 Radar level measurement 

Radar 1A level instruments measure the distance from the transmitter (located at some high point) 
to the surface of a process material located further below in much the same way as ultrasonic 
transmitters. The fundamental difference between a radar instrument and an ultrasonic instrument is 
the use of radio waves instead of sound waves. Radio waves are electromagnetic in nature (comprised 
of alternating electric and magnetic fields), and very high frequency (in the microwave frequency 
range - GHz). Sound waves are mechanical vibrations (transmitted from molecule to molecule in a 
fluid or solid substance) and of much lower frequency (tens or hundreds of kilohertz - still too high 
for a human being to detect as a tone) than radio waves. 

Some radar level instruments use waveguide "probes" to guide the radio waves into the process 
liquid while others send radio waves out through open space to reflect off the process material. 
The instruments using waveguides are called guided-wave radar instruments, whereas the radar 
instruments relying on open space for signal propagation are called non- contact radar. The 
differences between these two varieties of radar instruments is shown in the following illustration: 

Non-contact radar 
liquid level measurement 

Guided-wave radar 
liquid level measurement 



w Radio 
v , waves 

Probe ,, 

Non-contact radar transmitters are always mounted on the top side of a storage vessel. Modern 
radar transmitters are quite compact, as this photograph shows: 

14 "Radar" is an acronym: RAdio Detection And Ranging. First used as a method for detecting enemy ships and 
aircraft at long distances over the ocean in World War II, this technology is used for detecting the presence, distance, 
and/or speed of objects in a wide variety of applications. 



Probes used in guided-wave radar instruments may be single metal rods, parallel pairs of metal 
rods, or a coaxial metal rod-and-tube structure. Single-rod probes radiate the most energy, whereas 
coaxial probes do the best job guiding the microwave energy to the liquid interface and back. 
However, single- rod probes are much more tolerant of process fouling, where sticky masses of viscous 
liquid and/or solid matter cling to the probe. Such fouling deposits may cause radio energy reflections 
of sufficient magnitude to be misinterpreted by the radar instrument as a liquid level. 

Non-contact radar instruments rely on an antenna to direct microwave energy into the vessel, 
and to receive the echo (return) energy. These antennae must be kept clean and dry, which may be 
a problem if the liquid being measured emits condensible vapors. For this reason, non-contact radar 
instruments are often separated from the vessel interior by means of a dielectric window (made of 
some substance that is relatively "transparent" to radio waves yet acts as an effective vapor barrier) : 

13.5. ECHO 


Non-contact radar 
liquid level measurement 

Dielectric window 

Radio waves travel at the velocity of light 15 , 2.9979 x 10 8 meters per second in a perfect vacuum. 
The velocity of a radio wave through space depends on the dielectric permittivity (symbolized by 
the Greek letter "epsilon," e) of that space. A formula relating wave velocity to relative permittivity 
(the ratio of a substance's permittivity to that of a perfect vacuum, symbolized as e r and sometimes 
called the dielectric constant of the substance) and the velocity of light in a perfect vacuum (c) is 
shown here 16 : 

The relative permittivity of air at standard pressure and temperature is very nearly unity (1). 
The permittivity of any gas is a function of both pressure and temperature, as shown by the following 

1 + (e re f ~ 1) 






e r = Relative permittivity of a gas at a given pressure (P) and temperature (T) 

15 In actuality, both radio waves and light waves are electromagnetic in nature. The only difference between the 
two is frequency: while the radio waves used in radar systems are classified as "microwaves" with frequencies in the 
gigahertz (GHz) region, visible light waves range in the hundred of terahertz (THz)! 

16 This formula assumes lossless conditions: that none of the wave's energy is converted to heat while traveling 
through the dielectric. For many situations, this is true enough to assume. 



e re f = Relative permittivity of the same gas at standard pressure (P re f) and temperature (T re f) 

P = Absolute pressure of gas (bars) 

Pref = Absolute pressure of gas under standard conditions (~ 1 bar) 

T = Absolute temperature of gas (Kelvin) 

T re f = Absolute temperature of gas under standard conditions (« 273 K) 

If a radio wave encounters a sudden change in dielectric permittivity, some of that wave's energy 
will be reflected in the form of another wave traveling the opposite direction. In other words, the 
wave will "echo" when it reaches a discontinuity. This is the basis of all radar devices: 

Radar transceiver 

This same principle explains reflected signals in copper transmission lines as well. If anything 
happens along the length of a transmission line to cause a discontinuity (a sudden change in 
characteristic impedance), a portion of the signal's power will be reflected back to the source. In a 
transmission line, continuities may be formed by pinches, breaks, or short-circuits. In a radar level 
measurement system, any sudden change in permittivity is a discontinuity that will reflect some of 
the radio energy back to the source. 

The ratio of reflected power to incident (transmitted) power at any interface of materials is called 
the power reflection factor (R). This may be expressed as a unitless ratio, or more often as a decibel 
figure. The relationship between dielectric permittivity and reflection factor is as follows: 



{y^r~2 + V^y 


R = Power reflection factor at interface, as a unitless ratio 

e r i = Relative permittivity (dielectric constant) of the first medium 

13.5. ECHO 399 

e r 2 = Relative permittivity (dielectric constant) of the second medium 

The fraction of incident power transmitted through the interface {Pforward) is, of course, the 
mathematical complement of the power reflection factor: 1 — R. 

For situations where the first medium is air or some other low-permittivity gas, the formula 
simplifies to the following form (with e r being the relative permittivity of the reflecting substance) : 

— 2 


In the previous illustration, the two media were air (e r w 1) and water (e r w 80) - a nearly ideal 
scenario for strong signal reflection. Given these relative permittivity values, the power reflection 
factor has a value of 0.638 (63.8%), or -1.95 dB. This means that well over half the incident power 
gets reflected by the air/water interface, with the remaining 0.362 (36.2%) of the wave's power 
making it through the air-water interface and propagating into water. If the liquid in question is 
gasoline rather than water (having a rather low relative permittivity value of approximately 2), the 
power reflection ratio will only be 0.0294 (2.94%) or -15.3 dB, with the vast majority of the wave's 
power successfully penetrating the air-gasoline interface. 

The longer version of the power reflection factor formula suggests liquid-liquid interfaces should 
be detectable using radar, and indeed they are. All that is needed is a sufficiently large difference in 
relative permittivity between the two liquids to create a strong enough echo to reliably detect. 
Liquid-liquid interface level measurement with radar works best when the upper liquid has a 
substantially lesser permittivity value than the lower liquid 17 . A layer of hydrocarbon oil on top of 
water (or any aqueous solution such as an acid or a caustic) is a good candidate for guided-wave 
radar level measurement. An example of a liquid-liquid interface that would be very difficult for a 
radar instrument to detect is water (e r ~ 80) above glycerin (e r ~ 42). If the radar instrument uses 
a digital network protocol to communicate information with a host system (such as HART or any 
number of "fieldbus" standards), it may perform as a multi- variable transmitter, transmitting both 
the interface level measurement and the total liquid level measurement simultaneously. 

One reason why a lesser-e fluid above a greater-e fluid is easier to detect than the inverse is due 
to the necessity of the signal having to travel through a gas-liquid interface above the liquid-liquid 
interface. With gases and vapors having such small e values, the signal would have to pass through 
the gas-liquid interface first in order to reach the liquid-liquid interface. This gas-liquid interface, 
having the greatest difference in e values of any interface within the vessel, will be most reflective 
to radio energy in both directions. Thus, only a small portion of the incident wave will ever reach 
the liquid- liquid interface, and a similarly small portion of the wave reflected off the liquid- liquid 
interface (which itself is a fraction of the forward wave power that made it through the gas-liquid 
interface on its way down) will ever make it through the gas-liquid interface on its way back up to 
the instrument. The situation is much improved if the e values of the two liquid layers are inverted, 
as shown in this hypothetical comparison (all calculations 18 assume no power dissipation along the 
way, only reflection at the interfaces): 

17 Rosemount's "Replacing Displacers with Guided Wave Radar" technical note states that the difference in dielectric 
constant between the upper and lower liquids must be at least 10. 

!8_R = 0.5285 for the 1/40 interface; R = 0.02944 for the 40/80 interface; and R = 0.6382 for the 1/80 interface. 



Signal power strengths en route and 
reflected off of the liquid-liquid interface 




' 0.655% 


' 1.388% 


£ = 40 

£ = 80 


" 0.385% 





£ = 40 

As you can see in the illustration, the difference in power received back at the instrument is 
nearly two to one, just from the upper liquid having the lesser of two identical e values. Of course, 
in real life you do not have the luxury of choosing which liquid will go on top of the other, but you 
do have the luxury to choose the appropriate liquid-liquid interface level measurement technology, 
and as you can see here certain orientations of e values are less detectable with radar than others. 

Another factor working against radar as a liquid-liquid interface measurement technology for 
interfaces where the upper liquid has a greater dielectric constant is that fact that many high-e 
liquids are aqueous in nature, and water readily dissipates microwave energy. This fact is exploited 
in microwave ovens, where microwave radiation excites water molecules in the food, dissipating 
energy in the form of heat. For a radar-based level measurement system consisting of gas/ vapor 
over water over some other (heavier) liquid, the echo signal will be extremely weak because the 
signal must pass through the "lossy" water layer twice before it returns to the radar instrument. 

Radio energy losses are important to consider in radar level instrumentation, even when the 
detected interface is simply gas (or vapor) over liquid. The power reflection factor formula only 
predicts the ratio of reflected power to incident power at an interface of substances. Just because an 
air-water interface reflects 63.8% of the incident power does not mean 63.8% of the incident power 
will actually return to the transceiver antenna! Any dissipative losses between the transceiver and 
the interface(s) of concern will weaken the radio signal, to the point where it may become difficult 
to distinguish from noise. 

Another important factor in maximizing reflected power is the degree to which the microwaves 
spread out on their way to the liquid interface(s) and back to the transceiver. Guided-wave radar 
instruments receive a far greater percentage of their transmitted power than non-contact radar 
instruments because the metal rod(s) used to guide the microwave signal pulses help prevent the 
waves from spreading (and therefore weakening) throughout the liquids as they propagate. In other 
words, the metal rod(s) function as a transmission line to direct and focus the microwave energy, 
ensuring a straight path from the instrument into the liquid, and a straight echo return path from the 
liquid back to the instrument. This is why guided-wave radar is the only practical radar technology 
for measuring liquid-liquid interfaces. 

13.5. ECHO 401 

A critically important factor in accurate level measurement using radar instruments is that 
the relative permittivity of the upper substance(s) (all media between the radar instrument and 
the interface of interest) be accurately known. The reason for this is rooted in the dependence 
of electromagnetic wave propagation velocity to relative permittivity. Recalling the wave velocity 
formula shown earlier: 


V = Velocity of electromagnetic wave through a particular substance 
c = Velocity of light in a perfect vacuum (k3x 10 8 meters per second) 
e r = Relative permittivity (dielectric constant) of the substance 

In the case of a single-liquid application where nothing but gas or vapor exists above the liquid, 
the permittivity of that gas or vapor must be precisely known. In the case of a two-liquid interface 
with gas or vapor above, the relative permittivities of both gas and upper liquids must be accurately 
known in order to accurately measure the liquid-liquid interface. Changes in dielectric constant 
value of the medium or media through which the microwaves must travel and echo will cause the 
microwave radiation to propagate at different velocities. Since all radar measurement is based on 
time-of-flight through the media separating the radar transceiver from the echo interface, changes 
in wave velocity through this media will affect the amount of time required for the wave to travel 
from the transceiver to the echo interface, and reflect back to the transceiver. Therefore, changes in 
dielectric constant directly affect the accuracy of any radar level measurement. 

Factors influencing the dielectric constant of gases include pressure and temperature, which 
means the accuracy of a radar level instrument will vary as gas pressure and/or gas temperature vary! 
Whether or not this variation is substantial enough to consider for any application depends on the 
desired measurement accuracy and the degree of permittivity change from one pressure/temperature 
extreme to another. In no case should a radar instrument be considered for any level measurement 
application unless the dielectric constant value(s) of the upper media are precisely known. This is 
analogous to the dependence on liquid density that hydrostatic level instruments face. It is futile to 
attempt level measurement based on hydrostatic pressure if liquid density is unknown, and it is just 
as futile to attempt level measurement based on radar if the dielectric constants are unknown 19 . 

As with ultrasonic level instruments, radar level instruments have the ability to measure the 
level of solid substances in vessels (e.g. powders and granules). The same caveat of repose angle 
applicable to ultrasonic level measurement (see page 392), however, is a factor for radar measurement 
as well. When the particulate solid is not very dense (i.e. much air between particles), the dielectric 
constant may be rather low, making the material surface more difficult to detect. 

19 For vented-tank level measurement applications where air is the only substance above the point of interest, the 
relative permittivity is so close to a value of 1 that there is little need for further consideration on this point. Where 
the relative permittivity of fluids becomes a problem for radar is in high-pressure (non-air) gas applications and 
liquid-liquid interface applications, especially where the upper substance composition is subject to change. 


13.6 Laser level measurement 

The least-common form of echo-based level measurement is laser, which uses pulses of laser light 
reflected off the surface of a liquid to detect the liquid level. Perhaps the most limiting factor 
with laser measurement is the necessity of having a sufficiently reflective surface for the laser light 
to "echo" off of. Many liquids are not reflective enough for this to be a practical measurement 
technique, and the presence of dust or thick vapors in the space between the laser and the liquid 
will disperse the light, weakening the light signal and making the level more difficult to detect. 

However, lasers have been applied with great success in measuring distances between objects. 
Applications of this technology include motion control on large machines, where a laser points at a 
moving reflector, the laser's electronics calculating distance to the reflector based on the amount of 
time it takes for the laser "echo" to return. The advent of mass-produced, precision electronics has 
made this technology practical and affordable for many applications. At the time of this writing 
(2008), it is even possible for the average American consumer to purchase laser "tape measures" for 
use in building construction! 

13.7. WEIGHT 


13.7 Weight 

Weight-based level instruments sense process level in a vessel by directly measuring the weight of 
the vessel. If the vessel's empty weight (tare weight) is known, process weight becomes a simple 
calculation of total weight minus tare weight. Obviously, weight-based level sensors can measure 
both liquid and solid materials, and they have the benefit of providing inherently linear mass storage 
measurement 20 . Load cells (strain gauges bonded to a steel element of precisely known modulus) are 
typically the primary sensing element of choice for detecting vessel weight. As the vessel's weight 
changes, the load cells compress or relax on a microscopic scale, causing the strain gauges inside to 
change resistance. These small changes in electrical resistance become a direct indication of vessel 

The following photograph shows three bins, each one supported by pillars equipped with load 
cells near their bases: 

One very important caveat for weight-based level instruments is to isolate the vessel from any 
external mechanical stresses generated by pipes or machinery. The following illustration shows a 
typical installation for a weight-based measurement system, where all pipes attaching to the vessel do 

20 Regardless of the vessel's shape or internal structure, the measurement provided by a weight-sensing system is 
based on the true mass of the stored material. Unlike height-based level measurement technologies (float, ultrasonic, 
radar, etc.), no characterization will ever be necessary to convert a measurement of height into a measurement of 



so through flexible couplings, and the weight of the pipes themselves is borne by outside structures 
through pipe hangers: 







" cell 

cell " 



Stress relief is very important because any forces acting upon the storage vessel will be interpreted 
by the load cells as more or less material stored in the vessel. The only way to ensure that the load 
cell's measurement is a direct indication of material held inside the vessel is to ensure that no other 
forces act upon the vessel except the gravitational weight of the material. 

An interesting problem associated with load cell measurement of vessel weight arises if there 
are ever electric currents traveling through the load cell(s). This is not a normal state of affairs, 
but it can happen if maintenance workers incorrectly attach arc welding equipment to the support 
structure of the vessel, or if certain electrical equipment mounted on the vessel such as lights or 
motors develop ground faults. The electronic amplifier circuits interpreting a load cell's resistance 
will detect voltage drops created by such currents, interpreting them as changes in load cell resistance 
and therefore as changes in material level. Sufficiently large currents may even cause permanent 
damage to load cells, as is often the case when the currents in question are generated by arc welding 

A variation on this theme is the so-called hydraulic load cell which is a piston-and-cylinder 
mechanism designed to translate vessel weight directly into hydraulic (liquid) pressure. A normal 
pressure transmitter then measures the pressure developed by the load cell and reports it as material 
weight stored in the vessel. Hydraulic load cells completely bypass the electrical problems associated 

13.7. WEIGHT 405 

with resistive load cells, but are more difficult to network for the calculation of total weight (using 
multiple cells to measure the weight of a large vessel) . 



13.8 Capacitive 

Capacitive level instruments measure electrical capacitance of a conductive rod inserted vertically 
into a process vessel. As process level increases, capacitance increases between the rod and the vessel 
walls, causing the instrument to output a greater signal. 

The basic principle behind capacitive level instruments is the capacitance equation: 





C = Capacitance 

e = Permittivity of dielectric (insulating) material between plates 

A = Overlapping area of plates 

d = Distance separating plates 

The amount of capacitance exhibited between a metal rod inserted into the vessel and the metal 
walls of that vessel will vary only with changes in permittivity (e), area (A), or distance (d). Since 
A is constant (the interior surface area of the vessel is fixed, as is the area of the rod once installed), 
only changes in e or d can affect the probe's capacitance. 

Capacitive level probes come in two basic varieties: one for conductive liquids and one for non- 
conductive liquids. If the liquid in the vessel is conductive, it cannot be used as the dielectric 
(insulating) medium of a capacitor. Consequently, capacitive level probes designed for conductive 
liquids are coated with plastic or some other dielectric substance, so that the metal probe forms one 
plate of the capacitor and the conductive liquid forms the other: 


Metal vessel 


In this style of capacitive level probe, the variable is distance (d), since the conductive liquid 
essentially acts to bring the vessel wall electrically closer to the probe. This means total capacitance 



will be greatest when the vessel is full (effective distance d is at a minimum), and least when the 
vessel is empty. 

If the liquid is non-conductive, it may be used as the dielectric itself, with the metal wall of the 
storage vessel forming the second capacitor plate: 


Metal vessel 


In this style of capacitive level probe, the variable is permittivity (e), provided the liquid has 
a substantially greater permittivity than the vapor space above the liquid. This means total 
capacitance will be greatest when the vessel is full (average permittivity e is at a maximum), and 
least when the vessel is empty. 

Permittivity of the process substance is a critical variable in the non-conductive style of 
capacitance level probe, and so good accuracy may be obtained with this kind of instrument only 
if the process permittivity is accurately known. A clever way to ensure good level measurement 
accuracy when the process permittivity is not stable over time is to equip the instrument with a 
special compensating probe (sometimes called a composition probe) below the LRV point in the vessel 
that will always be submerged in liquid. Since this compensating probe is always immersed, and 
always experiences the same A and d dimensions, its capacitance is purely a function of the liquid's 
permittivity (e). This gives the instrument a way to continuously measure process permittivity, 
which it then uses to calculate level based on the capacitance of the main probe. The inclusion of 
a compensating probe to measure and compensate for changes in liquid permittivity is analogous 
to the inclusion of a third pressure transmitter in a hydrostatic tank expert system to continuously 
measure and compensate for liquid density. It is a way to correct for changes in the one remaining 
system variable that is not related to changes in liquid level. 

Capacitive level instruments may be used to measure the level of solids (powders and granules) 
in addition to liquids. In these applications, and solid substance is almost always non-conductive, 
and therefore the permittivity of the substance becomes a factor in measurement accuracy. This 


can be problematic, as moisture content variations in the solid may greatly affect permittivity, as 
can variations in granule size. They are not known for great accuracy, though, primarily due to 
sensitivity to changes in process permittivity and errors caused by stray capacitance in probe cables. 

13.9 Radiation 

Certain types of nuclear radiation easily penetrates the walls of industrial vessels, but is attenuated 
by traveling through the bulk of material stored within those vessels. By placing a radioactive source 
on one side of the vessel and measuring the radiation making it through to the other side of the 
vessel, an approximate indication of level within that vessel may be obtained. 

The three most common forms of nuclear radiation are alpha particles (a), beta particles (/3), 
and gamma rays (7). Alpha particles are helium nuclei (2 protons bound together with 2 neutrons) 
ejected at high velocity from the nuclei of certain decaying atoms. They are easy to detect, but have 
very little penetrating power and so are not used for industrial level measurement. Beta particles are 
electrons 21 ejected at high velocity from the nuclei of certain decaying atoms. Like alpha particles, 
though, they have little penetrating power and so are not used for industrial level measurement. 
Gamma rays, on the other hand, are electromagnetic in nature (like X-rays and light waves) and 
have great penetrating power. This form of radiation is the most common used in industrial level 

One of the most effective methods of shielding against gamma ray radiation is with very 
dense substances such as lead or concrete. This is why the source boxes holding gamma-emitting 
radioactive pellets are lined with lead, so that the radiation escapes only in the direction intended: 


These "sources" may be locked out for testing and maintenance by moving a lever that hinges 
a lead shutter over the "window" of the box. This lead shutter acts as an on/off switch for the 
radioactive source. The lever actuating the shutter typically has provisions for lockout /tagout so 
that a maintenance person may place a padlock on the lever and prevent anyone else from "turning 
on" the source during maintenance. 

The accuracy of radiation-based level instruments varies with the stability of process fluid density, 
vessel wall coating, source decay rates, and detector drift. Given these error variables and the 
additional need for NRC (Nuclear Regulatory Commission) licensing to operate such instruments at 
an industrial facility, radiation instruments are typically used where no other instrument can possibly 
function. Examples include the level measurement of highly corrosive or toxic process fluids where 
penetrations into the vessel must be minimized and where piping requirements make weight-based 
measurement impractical, as well as processes where the internal conditions of the vessel are too 
violent for any instrument to survive (e.g. delayed coking vessels in the oil refining industry). 

21 Beta particles are not orbital electrons, but rather than product of elementary particle decay in an atom's nucleus. 
These electrons are spontaneously generated and subsequently ejected from the nucleus of the atom. 



13.10 Level sensor accessories 

Disturbances in the liquid tend to complicate liquid level measurement. These disturbances may 
result from liquid introduced into a vessel above the liquid level (splashing into the liquid's surface), 
the rotation of agitator paddles, and/or turbulent flows from mixing pumps. Any source of 
turbulence for the liquid surface (or liquid-liquid interface) is especially problematic for echo-type 
level sensors, which only sense interfaces between vapors and liquids, or liquids and liquids. 

If it is not possible to eliminate disturbances inside the process vessel, a relatively simple accessory 
one may add to the process vessel is a vertical length of pipe called a stilling well. To understand 
the principle of a stilling well, first consider the application of a hydraulic oil reservoir where we 
wish to continuously measure oil level. The oil flow in and out of this reservoir will cause problems 
for the displacer element: 

"choppy" liquid surface 


\Turbulence will impose vertica 
oscillations on the displacer 

Oil flow will impose a 

lateral force on the displacer 


A section of vertical pipe installed in the reservoir around the displacer will serve as a shield to 
all the turbulence in the rest of the reservoir. The displacer element will no longer be subject to 
a horizontal blast of oil entering the reservoir, nor any wave action to make it bob up and down. 
This section of pipe quiets, or stills, the oil surrounding the displacer, making it easier to measure 
oil level: 

Flow 1^ 



Stilling wells may be used in conjunction with many types of level instruments: floats, displacers, 
ultrasonic, radar, and laser to name a few. If the process application necessitates liquid-liquid 
interface measurement, however, the stilling well must be properly installed to ensure the interface 
level inside the well match the interface levels in the rest of the vessel. Consider this example of 
using a stilling well in conjunction with a tape-and-float system for interface measurement: 



^ well 


Float / 

In the left-hand installation where the stilling well is completely submerged, the interface levels 
will always match. In the right-hand installation where the top of the stilling well extends above 
the total liquid level, however, the two levels may not always match: 



or . . . 

The problem here is analogous to what we see with sightglass-style level gauges: interfaces may 
be reliably indicated if and only if both ends of the sightglass are submerged (see page 350 for an 
illustrated description of the problem). 

If it is not possible or practical to ensure complete submersion of the stilling well, an alternative 
technique is to drill holes or cut slots in the well to allow interface levels to equalize inside and 
outside of the well tube: 



Stilling well 


Slots cut into 
stilling well tube 

Such equalization ports are commonly found as a standard design feature on coaxial probes for 
guided-wave radar level transmitters, where the outer tube of the coaxial transmission line acts as 
a sort of stilling well for the fluid. Coaxial probes are typically chosen for liquid-liquid interface 
radar measurement applications because they do the best job of preventing dispersion of the radio 
energy 22 , but the "stilling well" property of a coaxial probe practically necessitates these equalization 
ports to ensure the interface level within the probe always matches the interface level in the rest of 
the vessel. 

13.11 Process/instrument suitability 

22 So much of the incident power is lost as the radar signal partially reflects off the gas-liquid interface, then the 
liquid-liquid interface, then again through the gas-liquid interface on its return trip to the instrument that every care 
must be taken to ensure optimum received signal strength. While twin-lead probes have been applied in liquid-liquid 
interface measurement service, the coaxial probe design is still the best for maintaining radar signal integrity. 



"Autolevel" Application Note AN 01C22A01-01E, Yokogawa Electric Corporation, 2006. 

"Boiler Drum Level Transmitter Calibration", application data sheet 00800-0100-3055, Rosemount, 
Inc., Chanhassen, MN, 2001. 

Brumbi, Detlef, Fundamentals of Radar Technology for Level Gauging, 4th Edition, Krohne 
Messtechnik GmbH & Co. KG, Duisburg, Germany, 2003. 

"Bubble Tube Installations For Liquid Level, Density, and Interface Measurements" , document MI 
020-328, The Foxboro Company, Foxboro, MA, 1988. 

Fribance, Austin E., Industrial Instrumentation Fundamentals, McGraw-Hill Book Company, New 
York, NY, 1962. 

Kallen, Howard P., Handbook of Instrumentation and Controls, McGraw-Hill Book Company, Inc., 
New York, NY, 1961. 

"Level Measurement Technology: Radar", document 00816-0100-3209, revision AA, Rosemount, 
Inc., Chanhassen, MN, 1999. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

MacBeth, Michael, IAEA CANDU Instrumentation & Control Course, SNERDI, Shanghai, 1998. 

"Model 1151 Alphaline Pressure Transmitters", product manual 00809-0100-4360, revision AA, 
Rosemount, Inc., Chanhassen, MN, 1997. 

"Replacing Displacers with Guided Wave Radar", technical note 3300_2_02_CA, Rosemount, Inc., 
Chanhassen, MN, 2003. 


Chapter 14 

Continuous temperature 


Temperature is the measure of average molecular kinetic energy within a substance. The concept 
is easiest to understand for gases under low pressure, where gas molecules randomly shuffle about. 
The average kinetic (motional) energy of these gas molecules defines temperature for that quantity 
of gas. There is even a formula expressing the relationship between average kinetic energy (E/~) and 
temperature (T) for a monatomic (single-atom molecule) gas: 

E k - — 


Ek = Average kinetic energy of the gas molecules (joules) 
k = Boltzmann's constant (1.38 x 1CP 23 joules/Kelvin) 
T = Absolute temperature of gas (Kelvin) 

Thermal energy is a different concept: the quantity of total kinetic energy for this random 
molecular motion. If the average kinetic energy is defined as ^~, then the total kinetic energy for 
all the molecules in a monatomic gas must be this quantity times the total number of molecules (TV) 
in the gas sample: 

_ 3NkT 
thermal — 2 

This may be equivalently expressed in terms of the number of moles of gas rather than the 
number of molecules (a staggeringly large number for any realistic sample) : 





^thermal = Total thermal energy for a gas sample (joules) 

n = Quantity of gas in the sample (moles) 

R = Ideal gas constant (8.315 joules per mole-Kelvin) 



T = Absolute temperature of gas (Kelvin) 

Heat is denned as the exchange of thermal energy from one sample to another, by way of 
conduction (direct contact), convention (transfer via a moving fluid), or radiation (emitted energy); 
although you will often find the terms thermal energy and heat used interchangeably. 

Temperature is a more easily detected quantity than heat. There are many different ways to 
measure temperature, from a simple glass-bulb mercury thermometer to sophisticated infra-red 
optical sensor systems. Like all other areas of measurement, there is no single technology that 
is best for all applications. Each temperature-measurement technique has its own strengths and 
weaknesses. One responsibility of the instrument technician is to know these pros and cons so as 
to choose the best technology for the application, and this knowledge is best obtained through 
understanding the operational principles of each technology. 


14.1 Bi-metal temperature sensors 

Solids tend to expand when heated. The amount that a solid sample will expand with increased 
temperature depends on the size of the sample, the material it is made of, and the amount of 
temperature rise. The following formula relates linear expansion to temperature change: 

I = l (l+aAT) 


I = Length of material after heating 
Zo = Original length of material 
a = Coefficient of linear expansion 
AT = Change in temperature 

Here are some typical values of a for common metals: 

• Aluminum = 25 x 10 -6 per degree C 

• Copper = 16.6 x 10~ 6 per degree C 

• Iron = 12 x 10~ 6 per degree C 

• Tin = 20 x 1CP 6 per degree C 


Titanium = 8.5 x 10 per degree C 

As you can see, the values for a are quite small. This means the amount of expansion (or 
contraction) for modest temperature changes are almost too small to see unless the sample size (Iq) 
is huge. We can readily see the effects of thermal expansion in structures such as bridges, where 
expansion joints must be incorporated into the design to prevent serious problems due to changes 
in ambient temperature. However, for a sample the size of your hand the change in length from a 
cold day to a warm day will be microscopic. 

One way to amplify the motion resulting from thermal expansion is to bond two strips of dissimilar 
metals together, such as copper and iron. If we were to take two equally-sized strips of copper and 
iron, lay them side-by-side, and then heat both of them to a higher temperature, we would see the 
copper strip lengthen slightly more than the iron strip: 



-* I I iron 

If we bond these two strips of metal together, this differential growth will result in a bending 
motion that greatly exceeds the linear expansion. This device is called a bi-metal strip: 

Bending 1 copper 




This bending motion is significant enough to drive a pointer mechanism, activate an 
electromechanical switch, or perform any number of other mechanical tasks, making this a very 
simple and useful primary sensing element for temperature. 

If a bi-metallic strip is twisted over a long length, it will tend to un-twist as it heats up. This 
twisting motion may be used to directly drive the needle of a temperature gauge. This is the 
operating principle of the temperature gauge shown in the following photograph: 


14.2 Filled-bulb temperature sensors 


Filled-bulb systems exploit the principle of fluid expansion to measure temperature. If a fluid is 
enclosed in a sealed system and then heated, the molecules in that fluid will exert a greater pressure 
on the walls of the enclosing vessel. By measuring this pressure, and/or by allowing the fluid to 
expand under constant pressure, we may infer the temperature of the fluid. 

Class I and Class V systems use a liquid fill fluid (class V is mercury). Here, the volumetric 
expansion of the liquid drives an indicating mechanism to show temperature: 



Class I or Class V 

Class III systems use a gas fill fluid instead of liquid. Here, the change in pressure with 
temperature (as described by the Ideal Gas Law) allows us to sense the bulb's temperature: 




Pointer Scale 

In these systems, it is quite critical that the tube connecting the sensing bulb to the indicating 
element be of minimal volume, so that the fluid expansion is primarily due to changes in temperature 
at the bulb rather than changes in temperature along the length of the tube. It is also important 
to realize that the fluid volume contained by the bellows (or bourdon tube or diaphragm . . .) 
is also subject to expansion and contraction due to temperature changes at the indicator. This 
means the temperature indication varies somewhat as the indicator temperature changes, which is 
not desirable, since we intend the device to measure temperature (exclusively) at the bulb. Various 
methods of compensation exist for this effect (for example, a bi-metal spring inside the indicator 
mechanism to automatically offset the indication as ambient temperature changes), but it may be 
permanently offset through a simple "zero" adjustment provided that the ambient temperature at 
the indicator does not change much. 

A fundamentally different class of filled-bulb system is the Class II, which uses a volatile 
liquid/vapor combination to generate a temperature-dependent fluid expansion: 



Pointer Scale 

Pointer Scale 

Pointer Scale 



liquid " 

Nonvolatile ^^ 


Class MB 


Class IID 



Volatile liquid — 




• " 

Given that the liquid and vapor are in direct contact with each other, the pressure in the system 
will be precisely equal to the saturated vapor pressure at the vapor/liquid interface. This makes 
the Class II system sensitive to temperature only at the bulb and nowhere else along the system's 
volume. Because of this phenomenon, a Class II filled-bulb system requires no compensation for 
temperature changes at the indicator. 

Class II systems do have one notable idiosyncrasy, though: they have a tendency to switch from 
Class IIA to Class IIB when the temperature of the sensing bulb crosses the ambient temperature at 
the indicator. Simply put, the liquid tends to seek the colder portion of a Class II system while the 
vapor tends to seek the warmer portion. This causes problems when the indicator and sensing bulb 
exchange identities as warmer/colder. The rush of liquid up (or down) the capillary tubing as the 
system tries to reach a new equilibrium causes intermittent measurement errors. Class II filled-bulb 
systems designed to operate in either IIA or IIB mode are classified as IIC. 

One calibration problem common to all systems with liquid-filled capillary tubes is an offset in 
temperature measurement due to hydrostatic pressure (or suction) resulting from a different in height 
between the measurement bulb and the indicator. This represents a "zero" shift in calibration, which 
may be permanently offset by a "zero" adjustment at the time of installation. Class III (gas-filled) 
and Class IIB (vapor-filled) systems, of course, suffer no such problem because there is no liquid in 
the capillary tube to generate a pressure. 



14.3 Thermistors and Resistance Temperature Detectors 

One of the simplest classes of temperature sensor is one where temperature effects a change in 
electrical resistance. With this type of primary sensing element, a simple ohmmeter is able to 
function as a thermometer, interpreting the resistance as a temperature measurement: 


Thermistor /^\ 



Thermistors are devices made of metal oxide which either increase in resistance with increasing 
temperature (a positive temperature coefficient) or decrease in resistance with increasing temperature 
(a negative temperature coefficient). RTDs are devices made of pure metal (usually platinum 
or copper) which always increase in resistance with increasing temperature. The major different 
between thermistors and RTDs is linearity: thermistors are highly sensitive and nonlinear, whereas 
RTDs are relatively insensitive but very linear. For this reason, thermistors are typically used 
where high accuracy is unimportant. Many consumer-grade devices use thermistors for temperature 

Resistive Temperature Detectors (RTDs) relate resistance to temperature by the following 

R T = R ref [l + a(T - T ref )} 


Rt = Resistance of RTD at given temperature T (ohms) 

R re f = Resistance of RTD at the reference temperature T re f (ohms) 

a = Temperature coefficient of resistance (ohms per ohm/degree) 

Due to nonlinearities in the RTD's behavior, the above formula is only an approximation. A 
better approximation is the Callendar-van Dusen formula, which introduces second, third, and 
fourth-degree terms for a better fit: R T = R ref (l + AT + BT 2 - 100CT 3 + CT 4 ) for temperatures 
ranging -200° C < T < 0° C and R T = R ref {l + AT + BT 2 ) for temperatures ranging 0° C < T < 
661° C, both assuming T ref = 0° C. 

Water's melting/freezing point is the standard reference temperature for most RTDs. Here are 
some typical values of a for common metals: 

• Nickel = 0.00672 ft/ft°C 

• Tungsten = 0.0045 f2/Jl°C 

Silver = 0.0041 fi/fi°C 



• Gold = 0.0040 n/n°c 

• Platinum = 0.00392 Q/ft°C 

• Copper = 0.0038 fi/fi°C 

100 Q is a very common reference resistance (R re f) for industrial RTDs. 1000 Q is another 
common reference resistance. Compared to thermistors with their tens or even hundreds of thousands 
of ohms' nominal resistance, an RTD's resistance is comparatively small. This can cause problems 
with measurement, since the wires connecting an RTD to its ohmmeter possess their own resistance, 
which will be a more substantial percentage of the total circuit resistance than for a thermistor. 

The following schematic diagrams show the relative effects of 2 ohms total wire resistance on a 
thermistor circuit and on an RTD circuit: 

R ref = 50k Q 

K-wire — J " 


"wire — ^ " 



R tntnl = 50,002 n 




= ioo n 

K-wire — ^ " 


"wire — ^ " 



R mal = 102 Q 

Clearly, wire resistance is more problematic for low-resistance RTDs than for high-resistance 
thermistors. In the RTD circuit, wire resistance counts for 1.96% of the total circuit resistance. In 
the thermistor circuit, the same 2 ohms of wire resistance counts for only 0.004% of the total circuit 
resistance. The thermistor's huge reference resistance value "swamps" 1 the wire resistance to the 
point that the latter becomes insignificant by comparison. 

In HVAC (Heating, Ventilation, and Air Conditioning) systems, where the temperature 
measurement range is relatively narrow, the nonlinearity of thermistors is not a serious concern 
and their relative immunity to wire resistance is a decided advantage over RTDs. In industrial 
temperature measurement applications where the temperature ranges are usually much wider, the 

1 "Swamping" is the term given to the overshadowing of one effect by another. Here, the normal resistance of the 
high-value RTD greatly overshadows any wire resistance, such that wire resistance becomes negligible. 



nonlinearity of thermistors is a significant problem, so we must find a way to deal with the (lesser) 
problem of wire resistance. 

A very old electrical technique known as the Kelvin or four-wire method is a practical solution 
for this problem. Commonly employed to make precise resistance measurements for scientific 
experiments in laboratory conditions, the four-wire technique uses four wires to connect the 
resistance under test (in this case, the RTD) to the measuring instrument: 

R ref = won 

"wire — ^ " 

AV — 


AV — 

AV — 





Current is supplied to the RTD from a current source, whose job it is to precisely regulate 
current regardless of circuit resistance. A voltmeter measures the voltage dropped across the RTD, 
and Ohm's Law is used to calculate the resistance of the RTD (R = y). 

None of the wire resistances are consequential in this circuit. The two wires carrying current to the 
RTD will drop some voltage along their length, but this is of no concern because the voltmeter only 
"sees" the voltage dropped across the RTD. The two wires connecting the voltmeter to the RTD have 
resistance, but drop negligible voltage because the voltmeter draws so little current through them 
(remember an ideal voltmeters has infinite input impedance, and modern semiconductor-amplified 
voltmeters have impedances of several mega-ohms or more) . 

The only disadvantage of the four-wire method is the sheer number of wires necessary. Four 
wires per RTD can add up to a sizeable wire count when many different RTDs are involved on the 
same process. Wires cost money, and occupy expensive conduit, so there are situations where the 
four-wire method is a burden. 

A compromise between two-wire and four-wire RTD connections is the three-wire connection, 
which looks like this: 



R ref = 100 n 



Voltmeter B 


■"wire ~ -* ^ 





Voltmeter A 

In a three- wire RTD circuit, voltmeter "A" measures the voltage dropped across the RTD (plus 
the voltage dropped across the bottom current-carrying wire). Voltmeter "B" measures just the 
voltage dropped across the top current-carrying wire. Assuming both current-carrying wires will have 
(very nearly) the same resistance, subtracting the indication of voltmeter "B" from the indication 
given by voltmeter "A" yields the voltage dropped across the RTD. 

Vrtd = V 


meter(A) meter(B) 

Of course, real RTD instruments do not typically employ direct-indicating voltmeters. Most 
often, the voltage-measuring element is an analog-to-digital converter (ADC) which sends a digital 
output to a microprocessor for processing and signal output and/or display. Analog electronic RTD 
instruments have also been built, using operational amplifiers to convert the RTD's voltage drop into 
a standard instrument output signal, such as 4-20 mA DC. The voltmeters shown in the previous 
diagrams serve only to illustrate the basic concepts. 

One problem inherent to both thermistors and RTD's is self-heating. In order to measure the 
resistance of either device, we must pass an electric current through it. Unfortunately, this results 
in the generation of heat at the resistance according to Joule's Law: 

P = TR 

This dissipated power causes the thermistor or RTD to increase in temperature beyond its 
surrounding environment, introducing a positive measurement error. The effect may be minimized 
by limiting excitation current to a bare minimum, but this results in less voltage dropped across 
the device. The smaller the developed voltage, the more sensitive the voltage-measuring instrument 
must be to accurately sense the condition of the resistive element. Furthermore, a decreased signal 
voltage means we will have a decreased signal-to-noise ratio, all other factors being equal. 

One clever way to circumvent the self-heating problem without diminishing excitation current 
to the point of uselessness is to pulse current through the resistive sensor and digitally sample the 
voltage only during those brief time periods while the thermistor or RTD is powered. This technique 
works well when we are able to tolerate slow sample rates from our temperature instrument, which 


is often the case because most temperature measurement applications are slow-changing by nature. 
The pulsed-current technique enjoys the further advantage of reducing power consumption for the 
instrument, an important factor in battery-powered temperature measurement applications. 



14.4 Thermocouples 

When two dissimilar metal wires are joined together at one end, a voltage is produced at the other 
end that is approximately proportional to temperature. That is to say, the junction of two different 
metals behaves like a temperature-sensitive battery. This form of electrical temperature sensor is 
called a thermocouple: 

Iron (Fe) wire 


1 T" 



Copper (Cu) wire 


This phenomenon provides us with a simple and direct way to electrically infer temperature: 
simply measure the voltage produced by the junction, and you can tell the temperature of that 
junction. And it would be that simple, if it were not for an unavoidable consequence of electric 
circuits: when we connect any kind of electrical instrument the iron and copper wires, we inevitably 
produce another junction of dissimilar metals. The following schematic shows this fact. 










J 3 

Junction 3\ is a junction of iron and copper - two dissimilar metals - which will generate a voltage 
related to temperature. Note that junction J2, which is necessary for the simple fact that we must 
somehow connect our copper- wired voltmeter to the iron wire, is also a dissimilar-metal junction 
which will generate a voltage related to temperature. Note also how the polarity of junction J 2 
stands opposed to the polarity of junction J\ (iron = positive ; copper = negative). A third junction 
( J3) also exists between wires, but it is of no consequence because it is a junction of two identical 
metals which does not generate a temperature-dependent voltage at all. 

The presence of this second voltage-generating junction (J2) helps explain why the voltmeter 
registers volts when the entire system is at room temperature: any voltage generated by the iron- 
copper junctions will be equal in magnitude and opposite in polarity, resulting in a net (series-total) 
voltage of zero. It is only when the two junctions J\ and Ji are at different temperatures that the 
voltmeter registers any voltage at all. 



Thus, thermocouple systems are fundamentally differential temperature sensors. That is, they 
provide an electrical output proportional to the difference in temperature between two different 
points. For this reason, the wire junction we use to measure the temperature of interest is called the 
measurement junction while the other junction (which we cannot get rid of) is called the reference 

Multiple techniques exist to deal with the influence of the reference junction's temperature. 
One technique is to physically fix the temperature of that junction at some constant value so it is 
always stable. This way, any changes in measured voltage must be due to changes in temperature 
at the measurement junction, since the reference junction has been rendered incapable of changing 
temperature. This may be accomplished by immersing the reference junction in a bath of ice and 









In fact, this is how thermocouple temperature/voltage tables are referenced: describing the 
amount of voltage produced for given temperatures at the measurement junction with the reference 
junction held at the freezing point of water (0 °C = 32 °F). 

However, this is not a very practical solution for dealing with the reference junction's voltage. 
Instead, we could apply an additional electrical circuit to counter-act the voltage produced by the 
reference junction. This is called a reference junction compensation or cold junction compensation 

Compensating for the effects ofJ 2 
using a "reference junction compensation" 
circuit to generate a counter-voltage 








Please note that "cold junction" is just a synonymous label for "reference junction." In 
fact the "cold" reference junction may very well be at a warmer temperature than the so-called 



"hot" measurement junction! Nothing prevents anyone from using a thermocouple to measure 
temperatures below freezing. 

This compensating voltage source (V r j c in the above schematic) uses some other temperature- 
sensing device such as a thermistor or RTD to sense the local temperature at the terminal block 
where junction Ji is formed, and produce a counter-voltage that is precisely equal and opposite to 
J2's voltage. Having canceled the effect of the reference junction, the voltmeter now only registers 
the voltage produced by the measurement junction J\. 

At first it may seem pointless to go through the trouble of building a reference junction 
compensation circuit. After all, why bother to do this just to be able to use a thermocouple to 
accurately measure temperature, when we could simply use this "other" device (thermistor, RTD, 
etc.) to directly measure the temperature of interest? In other words, isn't the usefulness of a 
thermocouple invalidated if we have to go through the trouble of integrating another type of electrical 
temperature sensor in the circuit just to compensate for an idiosyncrasy of thermocouples? 

The answer to this very good question is that thermocouples enjoy certain advantages over 
these other sensor types. Thermocouples are extremely rugged and have far greater temperature- 
measurement ranges than thermistors, RTDs, and other primary sensing elements. However, if the 
application does not demand extreme ruggedness or large measurement ranges, a thermistor or RTD 
would likely be the better choice. 

Thermocouples exist in many different types, each with its own color codes for the dissimilar- 
metal wires. Here is a table showing the more common thermocouple types: 


Positive wire 

Negative wire 

Temperature range 


copper (blue) 

constantan (red) 

-300 to 700 °F 


iron (white) 

constantan (red) 

32 to 1400 °F 


chromel (violet) 

constantan (red) 

32 to 1600 °F 


chromel (yellow) 

alumel (red) 

32 to 2300 °F 


Pt90% - RhlO% (black) 

Platinum (red) 

32 to 2700 °F 


Pt70% - Rh30% (black) 

Pt94% - Rh6% (red) 

32 to 3380 °F 

It is critical to realize that the phenomenon of a "reference junction" is an inevitable effect of 
having to close the electric circuit loop in a circuit made of dissimilar metals. This is true regardless 
of the number of metals involved. In the last example, only two metals were involved: iron and 
copper. This formed one iron-copper junction (Ji) at the measurement end and one iron-copper 
junction ( Ji) at the indicator end. Recall that the copper-copper junction J3 was of no consequence 
because its identical metallic composition generates no thermal voltage: 











J 3 

The same thing happens when we form a thermocouple out of two metals, neither one being 
copper. Take for instance this example of a type J thermocouple: 









- + 

J 3 

Here we have three voltage-generating junctions: J\ of iron and constantan, Ji of iron and copper, 
and J3 of copper and constantan which just happens to be the metallic combination for a type T 
thermocouple. Upon first inspection it would seem we have a much more complex situation than we 
did with just two metals (iron and copper), but in actuality the situation is as simple as before. 

A principle we apply in thermocouple circuit analysis called the Law of Intermediate Metals helps 
us simplify the situation. According to this law, intermediate metals in a series of junctions are of 
no consequence to the overall (net) voltage so long as those intermediate junctions are all at the 
same temperature. Representing this pictorially, the net effect of having four different metals (A, B, 
C, and D) joined together in series is the same as just having the first and last metal in that series 
(A and D) joined with one junction, if all intermediate junctions are at the same temperature: 






If all junctions are at the same 
temperature, it is equivalent to . 

In our Type J thermocouple circuit where iron and constantan both join to copper, we see copper 
as an intermediate metal so long as junctions J2 and J3 are at the same temperature. Since those 
two junctions are located next to each other on the indicating instrument, identical temperature is a 
reasonable assumption, and we may treat junctions Ji and J3 as a single iron-constantan reference 
junction. In other words, the Law of Intermediate Metals tells us we can treat these two circuits 










- + 







(made of constantan wire) 



(no voltage) 


The practical import of all this is we can always treat the reference junction(s) as a single junction 
made from the same two metal types as the measurement junction, so long as all dissimilar metal 



junctions at the reference location are of equal temperature. 

This fact is extremely important in the age of semiconductor circuitry, where the connection of 
a thermocouple to an electronic amplifier involves many different junctions, from the thermocouple 
wires to the amplifier's silicon. Here we see a multitude of reference junctions, inevitably formed by 
the necessary connections from thermocouple wire to the silicon substrate inside the amplifier chip: 


brass copper iead/tin kovar gold 







Constantan brass copper iead/tin kovar gold 







It should be obvious that each complementary junction pair cancels if each pair is at the same 
temperature (e.g. gold-silicon junction J 12 cancels with silicon-gold junction J 13 because they 
generate the exact same amount of voltage with opposing polarities). The Law of Intermediate 
Metals goes one step further by telling us junctions J2 through J13 taken together in series are of 
the same effect as a single reference junction of iron and constantan. Automatic reference junction 
compensation is as simple as counter-acting the voltage produced by this equivalent iron- const ant an 
junction at whatever temperature junctions Ji through J13 happen to be at. 

Previously, it was suggested this automatic compensation could be accomplished by intentionally 
inserting a temperature-dependent voltage source in series with the circuit, oriented in such a way 
as to oppose the reference junction's voltage: 

Compensating for the effects ofJ 2 
using a "reference junction compensation" 
circuit to generate a counter-voltage 








This technique is known as hardware compensation. A stand-alone circuit designed to do this 
is sometimes called an ice point, because it electrically accomplishes the same thing as physically 
placing the reference junction(s) in a bath of ice-water. 



A more modern technique for reference junction compensation is called software compensation. 
This is applicable only where the indicating device is microprocessor-based, and where an additional 
analog input channel exists. 


Compensating for the effects ofJ 2 
using a second input channel to sense 
ambient temperature and correcting 
mathematically in the computer 

Junction Copper 

+ J >- 









4-20 mA 

Instead of canceling the effect of the reference junction electrically, we can cancel the effect 
mathematically inside the microprocessor. Perhaps the greatest advantage of software compensation 
is flexibility. Being able to re-program the compensation function means we may use this device with 
different thermocouple types. With hardware-based compensation (an "ice point" circuit), re- wiring 
or replacement is necessary to accommodate different thermocouple types. 

Another consideration for thermocouples is burnout detection. The most common failure mode 
for thermocouples is to fail open, otherwise known as "burning out." An open thermocouple is 
problematic for any voltage-measuring instrument with high input impedance because the lack of a 
complete circuit on the input makes it possible for electrical noise from surrounding sources (power 
lines, electric motors, variable-frequency motor drives) to be detected by the instrument and falsely 
interpreted as a wildly varying temperature. 

For this reason it is prudent to design into the thermocouple instrument some provision for 
generating a consistent state in the absence of a complete circuit. This is called the burnout mode 
of a thermocouple instrument. 


switch rj] 




The resistor in this circuit provides a path for current in the event of an open thermocouple. 


It is sized in the mega-ohm range so that its effect is minimal during normal operation when the 
thermocouple circuit is complete. Only when the thermocouple fails open will the miniscule current 
through the resistor have any substantial effect on the voltmeter's indication. The SPDT switch 
provides a selectable burnout mode: in the event of a burnt-out thermocouple, we can configure the 
meter to either read high temperature (sourced by the instrument's internal milli-voltage source) or 
low temperature (grounded), depending on what failure mode we deem safest for the application. 


14.5 Optical temperature sensing 

Virtually any mass above absolute zero temperature will emit electromagnetic radiation (photons, 
or light) as a function of that temperature. The Stefan- Boltzmann Law of radiated energy tells us 
that the rate of heat lost by radiant emission from a hot object is proportional to the fourth power 
of the absolute temperature: 

** = eoAT^ 


-g- = Radiant heat loss rate (watts) 

e = Emissivity factor (unitless) 

a = Stefan-Boltzmann constant (5.67 x 10~ 8 W / m 2 • K 4 ) 

A = Surface area (square meters) 

T = Absolute temperature (Kelvin) 

This phenomenon provides us a way to infer an object's absolute temperature by sensing the 
radiation it emits. Such a measurement technique holds obvious advantages, perhaps the greatest 
being the lack of need for direct contact to the process with a sensing element such as an RTD or 

Using an array of radiation sensors it is possible to build a thermal imager, providing a graphic 
display of objects in its view according to their temperatures. Each object is artificially colored in 
the display on a chromatic scale that varies with temperature, hot objects typically registering as 
red tones and cold objects typically registering as blue tones. Thermal imaging is very useful in 
the electric power distribution industry, where technicians can check power line insulators and other 
objects at elevated potential for "hot spots" without having to make physical contact with those 
objects. Thermal imaging is also useful in performing "energy audits" of buildings and other heated 
structures, providing a means of revealing points of heat escape through walls, windows, and roofs. 

Perhaps the main disadvantage of optical temperature sensors is their inaccuracy. The emissivity 
factor (e) in the Stefan-Boltzmann equation varies with the composition of a substance, but beyond 
that there are several other factors (surface finish, shape, etc.) that affect the amount of radiation 
an optical sensor will receive from an object. For this reason, emissivity is not a very practical way 
to gauge the effectiveness of an optical temperature sensor. Instead, a more comprehensive measure 
of an object's "thermal-optical measureability" is emittance. 

A perfect emitter of thermal radiation is known as a blackbody. Emittance for a blackbody is 
unity (1), while emittance figures for any real object is a value between 1 and 0. The only certain way 
to know the emittance of an object is to test that object's thermal radiation at a known temperature. 
This assumes we have the ability to measure that object's temperature by direct contact, which of 
course renders void one of the major purposes of optical thermometry: to be able to measure an 
object's temperature without having to touch it. Not all hope is lost for optical techniques, though. 
All we have to do is obtain an emittance value for that object one time, and then we may calibrate 
an optical temperature sensor for that object's particular emittance so as to measure its temperature 
in the future without contact. 

Beyond the issue of emittance, other idiosyncrasies plague optical techniques as well. Objects 
also have the ability to reflect and transmit radiation from other bodies, which taints the accuracy 
of any optical device sensing the radiation from that body. An example of the former is trying to 


measure the temperature of a silver mirror using an optical pyrometer: the radiation received by the 
pyrometer is mostly from other objects, merely reflected by the mirror. An example of the latter is 
trying to measure the temperature of a gas or a clear liquid, and instead primarily measuring the 
temperature of a solid object in the background (through the gas or liquid). 

Nevertheless, optical techniques for measuring temperature have been and will continue to be 
useful in specific applications where other, contact-based techniques are impractical. 



14.6 Temperature sensor accessories 

One of the most important accessories for any temperature-sensing element is a pressure-tight sheath 
known as a thermowell. This may be thought of as a thermally conductive protrusion into a process 
vessel or pipe that allows a temperature-sensitive instrument to detect process temperature without 
opening a hole in the vessel or pipe. Thermowells are critically important for installations where 
the temperature element (RTD, thermocouple, thermometer, etc.) must be replaceable without 
de-pressurizing the process. 

Thermowells may be made out of any material that is thermally conductive, pressure-tight, and 
not chemically reactive with the process. A simple diagram showing a thermowell in use with a 
temperature gauge is shown here: 



Pipe wall 



Pipe wall 

If the temperature gauge is removed for service or replacement, the thermowell maintains pressure 
integrity of the pipe (no process fluid leaking out, and no air leaking in): 



(removed from process) 

Pipe wall 


Process fluid 

Pipe wall 



Pipe wall 

Photographs of a real (stainless steel) thermowell are shown here, the left-hand photo showing the 
entire length of the thermowell, and the right-hand photo showing the end where the temperature- 
sensing device is inserted: 

A photo of a complete RTD assembly (connection head, RTD, and thermowell) appears in the 



next photograph: 

Another photo shows an RTD installed in a thermowell on the side of a commercial freezer, using 
a Rosemount model 3044C temperature transmitter to output a 4-20 mA signal to an operator 



As useful as thermowells are, they are not without their caveats. First and foremost is the first- 
order time lag they add to the temperature measurement system by virtue of their mass and specific 
heat value. It should be intuitively obvious that one or more pounds of metal will not heat up and 
cool down as fast as a few ounces' worth of RTD or thermocouple, and therefore that the presence 
of a thermowell will decrease the response time of any temperature-sensing element. 

A potential problem with thermowells is incorrect installation of the temperature-sensing element. 
The element must be inserted with full contact at the bottom of the thermowell's blind hole. If any air 
gap is allows to exist between the end of the temperature element and the bottom of the thermowell's 
hole, this will add a second time lag to the measurement system 2 . Some thermowells include a spring 
clip in the bottom of the blind hole to help maintain constant metal-to-metal contact between the 
sensing element and the thermowell wall. 

2 The air gap acts as a thermal resistance while the mass of the element itself acts as a thermal capacitance. Thus, 
the inclusion of an air gap forms a thermal "RC time constant" delay network secondary to the thermal delay incurred 
by the thermowell. 


14.7 Process/instrument suitability 

The primary consideration for selecting a proper temperature sensing element for any application is 
the expected temperature range. Mechanical (bi-metal) and filled-system temperature sensors are 
limited to relatively low process temperatures, and cannot relay signals very far from the point of 

Thermocouples are by far the most rugged and wide-ranging of the contact-type temperature 
sensors. Accuracies vary with thermocouple type and installation quality. 

RTDs are more fragile than thermocouples, but they require no reference compensation and are 
inherently more linear. 

Optical sensors lack the ability to measure temperature of fluids inside vessels unless a transparent 
window is provided in the vessel for light emissions to reach the sensor. Otherwise, the best an optical 
sensor can do is report the skin temperature of a vessel. For monitoring surface temperatures of solid 
objects, especially objects that would be impractical or even dangerous to contact (e.g. electrical 
insulators on high- voltage power lines), optical sensors are the only appropriate solution. 

Chemical reactivity is a concern for contact- type sensors. If the sensing element is held inside 
a thermowell, that thermowell must be selected for minimum reaction with the process fiuid(s). 
Bare thermocouples are particularly vulnerable to chemical reactions given the nature of most 
thermocouple metals (iron, nickel, copper, etc.), and must be carefully chosen for the particular 
process chemistry to avoid reliability problems later. 


Beckerath, Alexander von; Eberlein, Anselm; Julien, Hermann; Kersten, Peter; and Kreutzer, 
Jochem, WIKA-Handbook, Pressure and Temperature Measurement, WIKA Alexander Wiegand 
GmbH & Co., Klingenberg, Germany, 1995. 

Fribance, Austin E., Industrial Instrumentation Fundamentals, McGraw-Hill Book Company, New 
York, NY, 1962. 

Irwin, J. David, The Industrial Electronics Handbook, CRC Press, Boca Raton, FL, 1997. 

Kallen, Howard P., Handbook of Instrumentation and Controls, McGraw-Hill Book Company, Inc., 
New York, NY, 1961. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 


Chapter 15 

Continuous fluid flow measurement 

Fluid flow may be measured volumetrically or by mass. Volumetric flow is expressed in volume 
units (e.g. gallons, liters, cubic inches) per unit time. Mass flow is expressed in mass units (slugs, 
kilograms, pounds-mass) per unit time. 

Liquids are essentially incompressible: that is, they do not easily yield in volume to applied 
pressure. Gases and vapors, however, easily change volume under the influence of changing pressure. 
In other words, a gas will yield to an increasing pressure by decreasing in volume as the gas molecules 
are forced closer together. This makes volumetric flow measurement more complex for gases than for 
liquids. To begin, we must agree on how to standardize volumetric measurement for a gas, when the 
volume is so easily subject to change. We can do this by agreeing on standard "base-line" pressures 
and temperatures under which a particular gas volume is specified. 

For example, a gas flow rate of 900 SCFM (Standard Cubic Feet per Minute) refers to 900 cubic 
feet of gas flowing per minute of time, if that gas flowstream were subjected to atmospheric pressure 
at 70° F (British units). The actual volume of gas moving through that pipe each minute under 
pressurized conditions will likely occupy far less than 900 cubic feet, due to physical compression 
(reduction of volume resulting from increased pressure) of the gas. However, the unit of "standard" 
cubic feet per minute gives people a common frame of reference. 



15.1 Pressure-based flowmeters 

All masses require force to accelerate (we can also think of this in terms of the mass generating a 
reaction force as a result of being accelerated). This is quantitatively expressed by Newton's Second 
Law of Motion: 





■ -■ — »> 


Newton's Second Law formula 
F - ma 

All fluids possess mass, and therefore require force to accelerate just like solid masses. If we 
consider a quantity of fluid confined inside a pipe 1 , with that fluid quantity having a mass equal to 
its volume multiplied by its mass density (m = pV, where p is the fluid's mass per unit volume), the 
force required to accelerate that fluid "plug" would be calculated just the same as for a solid mass: 

A volume of fluid 



Force (F) 

■■-— ►) 

Acceleration (a) 


(m - pV) 

Newton's Second Law formula 
F - ma F — pVa 

Since this accelerating force is applied on the cross-sectional area of the fluid plug, we may express 
it as a pressure, the definition of pressure being force per unit area: 




1 Sometimes referred to as a plug of fluid. 



P = P -a 

Since the rules of algebra required we divide both sides of the force equation by area, it left us 
with a fraction of volume over area (^) on the right-hand side of the equation. This fraction has a 
physical meaning, since we know the volume of a cylinder divided by the area of its circular face is 
simply the length of that cylinder: 


P = P -a 

P = pla 

When we apply this to the illustration of the fluid mass, it makes sense: the pressure described 
by the equation is actually a differential 2 pressure drop from one side of the fluid mass to the other, 
with the length variable (/) describing the spacing between the differential pressure ports: 

Length (I) 
h- -H Pipe 



(m = pV)\ \ Acceleration (a) J 

— i — 
Pressure drop 


This tells us we can accelerate a "plug" of fluid by applying a difference of pressure across its 
length. The amount of pressure we apply will be in direct proportion to the density of the fluid and 
its rate of acceleration. Conversely, we may measure a fluid's rate of acceleration by measuring the 
pressure developed across a distance over which it accelerates. 

We may easily force a fluid to accelerate by altering its natural flow path. The difference of 
pressure generated by this acceleration will indirectly indicate the rate of acceleration. Since the 
acceleration we see from a change in flow path is a direct function of how fast the fluid was originally 
moving, the acceleration (and therefore the pressure drop) indirectly indicates fluid flow rate. 

A very common way to cause linear acceleration in a moving fluid is to pass the fluid through a 
constriction in the pipe, thereby increasing its velocity (remember that the definition of acceleration 
is a change in velocity). The following illustrations show several devices used to linearly accelerate 
moving fluids when placed in pipes, with differential pressure transmitters connected to measure the 
pressure drop resulting from this acceleration: 

2 What really matters in Newton's Second Law equation is the resultant force causing the acceleration. This is the 
vector sum of all forces acting on the mass. Likewise, what really matters in this scenario is the resultant pressure 
acting on the fluid plug, and this resultant pressure is the difference of pressure between one face of the plug and the 
other, since those two pressures will be acting in direct opposition to each other. 



Venturi tube 

Flow nozzle 




jjn- — stud 







Orifice plate 

Segmental wedge 

Another way we may accelerate a fluid is to force it to turn a corner through a pipe fitting called 



an elbow. This will generate radial acceleration, causing a pressure difference between the outside 
and inside of the elbow which may be measured by a differential pressure transmitter: 

Pipe elbow 

The pressure tap located on the outside of the elbow's turn registers a greater pressure than 
the tap located on the inside of the elbow's turn, due to the inertial force of the fluid's mass being 
"flung" to the outside of the turn as it rounds the corner. 

Yet another way to cause fluid acceleration is to force it to decelerate by bringing a portion of it 
to a full stop. The pressure generated by this deceleration (called the stagnation pressure) tells us 
how fast it was originally flowing. A few devices working on this principle are shown here: 




Pitot tube 


Pipe wall 


Pipe wall 





Pipe wall 


Pipe wall 

Pipe wall 

Drag disk 


Pipe wall 


pitot tube 


Pipe wall 

Holes - 

\ Flow 

Pipe wall 


Pipe wall 

\ Flow 

The following subsections in this flow measurement chapter explore different primary sensing 
elements (PSE's) used to generate differential pressure in a moving fluid stream. Despite their very 
different designs, they all operate on the same fundamental principle: causing a fluid to accelerate 
(or decelerate) by changing its flow path, and thus generating a measurable pressure difference. The 
following subsection will introduce a device called a venturi tube used to measure fluid flow rates, and 
derive mathematical relationships between fluid pressure and flow rate starting from basic physical 
conservation laws. 



15.1.1 Venturi tubes and basic principles 

The standard example used to demonstrate pressure change in a fluid stream is the venturi tube: a 
pipe purposefully narrowed to create a region of low pressure. If the fluid going through the venturi 
tube is a liquid under relatively low pressure, we may vividly show the pressure at different points 
in the tube by means of piezometers, which are transparent tubes allowing us to view liquid column 
heights. The greater the height of liquid column in the piezometer, the greater the pressure at that 
point in the flowstream: 


Point 1 

Point 2 

Point 3 

Ground level 


As indicated by the piezometer liquid heights, pressure at the constriction (point 2) is the least, 
while pressures at the wide portions of the venturi tube (points 1 and 3) are the greatest. This is a 
counter- intuitive result, but it has a firm grounding in the physics of mass and energy conservation. 
If we assume no energy is added (by a pump) or lost (due to friction) as fluid travels through this 
pipe, then the Law of Energy Conservation describes a situation where the fluid's energy must remain 
constant at all points in the pipe as it travels through. If we assume no fluid joins this flowstream 
from another pipe, or is lost from this pipe through any leaks, then the Law of Mass Conservation 
describes a situation where the fluid's mass flow rate must remain constant at all points in the pipe 
as it travels through. 

So long as fluid density remains fairly constant 3 , fluid velocity must increase as the cross-sectional 
area of the pipe decreases, as described by the Law of Continuity (see section 1.8.10 on page 53 for 
more details on this concept): 

Aivi = A 2 v 2 

3 This is a very sound assumption for liquids, and a fair assumption for gases when pressure changes through the 
venturi tube are modest. 


Rearranging variables in this equation to place velocities in terms of areas, we get the following 

V2 _ M 
vi A 2 

This equation tells us that the ratio of fluid velocity between the narrow throat (point 2) and the 
wide mouth (point 1) of the pipe is the same ratio as the mouth's area to the throat's area. So, if 
the mouth of the pipe had an area 5 times as great as the area of the throat, then we would expect 
the fluid velocity in the throat to be 5 times as great as the velocity at the mouth. Simply put, the 
narrow throat causes the fluid to accelerate from a lower velocity to a higher velocity. 

We know from our study of energy in physics that kinetic energy is proportional to the square 
of a mass's velocity (E^ = ^mv 2 ). If we know the fluid molecules increase velocity as they travel 
through the venturi tube's throat, we may safely conclude that those molecules' kinetic energies must 
increase as well. However, we also know that the total energy at any point in the fluid stream must 
remain constant, because no energy is added to or taken away from the stream in this simple fluid 
system. Therefore, if kinetic energy increases at the throat, potential energy must correspondingly 
decrease to keep the total amount of energy constant at any point in the fluid. 

Potential energy may be manifest as height above ground, or as pressure in a fluid system. Since 
this venturi tube is level with the ground, there cannot be a height change to account for a change in 
potential energy. Therefore, there must be a change of pressure (P) as the fluid travels through the 
venturi throat. The Laws of Mass and Energy Conservation invariably lead us to this conclusion: 
fluid pressure must decrease as it travels through the narrow throat of the venturi tube 4 . 

Conservation of energy at different points in a fluid stream is neatly expressed in Bernoulli's 
Equation as a constant sum of elevation, pressure, and velocity "heads" (see section 1.8.12 on page 
55 for more details on this concept): 

zipg +^f+Pi = z 2 pg +^ + P 2 


z = Height of fluid (from a common reference point, usually ground level) 
p = Mass density of fluid 
g = Acceleration of gravity 
V = Velocity of fluid 
P = Pressure of fluid 

We will use Bernoulli's equation to develop a precise mathematical relationship between pressure 
and flow rate in a venturi tube. To simplify our task, we will hold to the following assumptions for 
our venturi tube system: 

• No energy lost or gained in the venturi tube (all energy is conserved) 

• No mass lost or gained in the venturi tube (all mass is conserved) 

• Fluid is incompressible 

4 To see a graphical relationship between fluid acceleration and fluid pressures in a venturi tube, examine the 
illustration found on page 627. 


• Venturi tube centerline is level (no height changes to consider) 

Applying the last two assumptions to Bernoulli's equation, we see that the "elevation head" term 
drops out of both sides, since z, p, and g are equal at all points in the system: 

2 2 

Now we will algebraically re-arrange this equation to show pressures at points 1 and 2 in terms 
of velocities at points 1 and 2: 

v JP_ v ll = P _ P 
2 2 l 2 

Factoring | out of the velocity head terms: 

P_ (v 2_ vl)=Pl _p 2 

The Continuity equation shows us the relationship between velocities zq and v 2 and the areas at 
those points in the venturi tube, assuming constant density (p): 

A\v\ = A 2 v 2 

Specifically, we need to re-arrange this equation to define Vi in terms of vi so we may substitute 
into Bernoulli's equation: 

Vl = {t) V2 

Performing the algebraic substitution: 

P I 2 

Al ' V > 


Distributing the "square" power: 

5«" (£)'•*-«-* 

Factoring v\ out of the outer parentheses set: 

Solving for V2, step by step: 

¥<* -(£)">-*-* 

P4 -" ' -^ I (ft -ft) 


pv 2 2 = 2 | j—^ | (Pi ~ P2 




Pi -Pi 

Pi -Pi 

The result shows us how to solve for fluid velocity at the venturi throat (v 2 ) based on a difference 
of pressure measured between the mouth and the throat {P\ — Pi)- We are only one step away from 
a volumetric flow equation here, and that is to convert velocity (v) into flow rate (Q). Velocity is 
expressed in units of length per time (feet or meters per second or minute) , while volumetric flow 
is expressed in units of volume per time (cubic feet or cubic meters per second or minute). Simply 
multiplying throat velocity (V2) by throat area (A 2 ) will give us the result we seek: 

General flow/area/velocity relationship: 

Equation for throat velocity: 

v 2 = V2 

Q = Av 

1 P1-P2 



Multiplying both sides of the equation by throat area: 

1- . 1 P1-P2 

A 2 v 2 = V2A 2 


Now we have an equation solving for volumetric flow: 

Q = V2A : 

1 _ I A2. 

' ' A 1 


Please note how many constants we have in this equation. For any given venturi tube, the mouth 
and throat areas (Ai and A 2 ) will be fixed. This means the majority of this rather long equation 
are constant for any particular venturi tube, and therefore do not change with pressure, density, or 
flow rate. Knowing this, we may re-write the equation as a simple proportionality: 


'Pi- Pi 



To make this a more precise mathematical statement, we may insert a constant of proportionality 
(k) and once more have a true equation to work with: 

Q = J P >- P > 

The value of k depends, of course, on the physical dimensions of the venturi mouth and throat. 
A practical advantage to using a constant of proportionality is that we may adjust the value of k as 
necessary to account for factors other than just venturi tube geometry. One very important factor 
to consider is units of measurement. If the value of k is determined strictly by tube geometry, then 
the units used to express volumetric flow rate must correspond to the units used to express pressures 
and fluid density. For example, Q will be in units of cubic feet per second only if we insert pressure 
values Pi and Pi in units of pounds per square foot and mass density in units of slugs per cubic 
feet! If we wish to use more convenient units of measurement such as inches of water column for 
pressure and specific gravity (unitless) for density, the volumetric flow value produced by the raw 
equation will not be in any useful unit. 

However, if we know the differential pressure produced by any particular venturi tube with any 
particular fluid density at a specified flow rate, we may calculate the k value necessary to characterize 
that venturi tube for any other condition using those units. For example, if we know a particular 
venturi tube develops 45 inches of water column differential pressure at a flow rate of 180 gallons 
per minute of water (specific gravity = 1), we may plug these values into the equation and solve for 

Q = k J*-* 

180 = k 




Now that we know a value of 26.83 for k will yield gallons per minute of liquid flow through 
this venturi tube given pressure in inches of water column and density as a specific gravity, we may 
readily predict the flow rate through this tube for any other pressure drop we might happen to 




Specific gravity 

60 inches of water column differential pressure generated by a flow of water (specific gravity 
1) in this particular venturi tube gives us the following flow rate: 



Q = 26. 83 1 

Q = 207.8 GPM 

110 inches of water column differential pressure generated by a flow of gasoline (specific gravity 
0.657) in this same venturi tube gives us the following flow rate: 

Q = 26.83 


Q = 347 GPM 

If we wish to calculate mass flow instead of volumetric flow, the equation does not change much. 
The relationship between volume (V) and mass (to) for a sample of fluid is its mass density (p): 


P= V 

Similarly, the relationship between a volumetric flow rate (Q) and a mass flow rate (W) is also 
the fluid's mass density (p): 


P= Q 

Solving for W in this equation leads us to a product of volumetric flow rate and mass density: 

W = pQ 

A quick dimensional analysis check using common metric units confirms this fact. A mass flow 
rate in kilograms per second will be obtained by multiplying a mass density in kilograms per cubic 
meter by a volumetric flow rate in cubic meters per second: 





"m 3 " 

m 3 


Therefore, all we have to do to turn our general volumetric flow equation into a mass flow equation 
is multiply both sides by fluid density (p): 

Q = k< 

pQ = kp 

W = kp 



1 p 

J p ' 


V o 




It is generally considered "inelegant" to show the same variable more than once in an equation 
if it is not necessary, so let's try to consolidate the two densities (/?) using algebra. First, we may 
write p as the product of two square-roots: 


Next, we will break up the last radical into a quotient of two separate square roots: 

ur 7 r- /-V^i ~~ P% 
W = ky/p^/p 


Now we see how one of the square-rooted p terms cancels out the one in the denominator of the 

W = k^~p^P 1 - P 2 
Re-writing the two roots as one: 

W = k^/p{P 1 - P 2 ) 

As with the volumetric flow equation, all we need in order to arrive at a suitable k value for 
any particular venturi tube is a set of values taken from a real venturi tube in service, expressed 
in whatever units of measurement we desire. For example, if we had a venturi tube generating a 
differential pressure of 2.30 kilo-Pascals (kPa) at a mass flow rate of 500 kilograms per minute of 
naphtha (a petroleum product having a density of 0.665 kilograms per liter), we could solve for the 
k value of this venturi tube as such: 

W = k^p{P 1 - P 2 ) 
500 = *V(0.665)(2.3) 


fc = 404 

Now that we know a value of 404 for k will yield kilograms per minute of liquid flow through 
this venturi tube given pressure in kPa and density in kilograms per liter, we may readily predict 
the mass flow rate through this tube for any other pressure drop we might happen to measure: 






6.1 kPa of differential pressure generated by a flow of sea water (density = 1.03 kilograms per 
liter) in this particular venturi tube gives us the following mass flow rate: 

W = 404v / (1.03)(6.1) 



W = 1012 


It should be apparent by now that the relationship between flow rate (whether it be volumetric 
or mass) and differential pressure is non-linear: a doubling of flow rate will not result in a doubling 
of differential pressure. Rather, a doubling of flow rate will result in a quadrupling of differential 

This quadratic relationship between flow and pressure drop due to fluid acceleration requires 
us to mathematically "condition" or "characterize" the pressure signal sensed by the differential 
pressure instrument in order to arrive at an expressed value for flow rate. The customary solution 
to this problem is to incorporate a "square root" function between the transmitter and the flow 
indicator, as shown in the following diagram: 


pressure _, , . Indicating 

instrument Charactenzer gauge 

Direction of flow 

In the days of pneumatic instrumentation, this square-root function was performed in a separate 
device called a square root extractor. The Foxboro corporation model 557 pneumatic square root 
extractor was a classic example of this technology 5 : 

5 Despite the impressive craftsmanship and engineering that went into the design of pneumatic square root 
extractors, their obsolescence is mourned by no one. These devices were notoriously difficult to set up and calibrate 
accurately, especially as they aged. 



The modern solution is to incorporate digital square-root computation either in the indicator or 
in the transmitter itself. 



15.1.2 Orifice plates 

Of all the pressure-based flow elements in existence, the most common is the orifice plate. This is 
simply a metal plate with a hole in the middle for fluid to flow through. Orifice plates are typically 
sandwiched between two flanges of a pipe joint, allowing for easy installation and removal: 

Orifice plate 

Ol c3 vena contracta -sr^ _^_^ 
I ^j — . (point of maximum constriction) n — J 7* J ^ — v 

° o o o o ° 

Pipe wall 






The point where the fluid flow profile constricts to a minimum cross-sectional area after flowing 
through the orifice is called the vena contracta, and it is the area of minimum fluid pressure. The 
vena contracta corresponds to the narrow throat of a venturi tube. 

The simplest design of orifice plate is the square-edged, concentric orifice. This type of orifice 
plate is manufactured by machining a precise, straight hole in the middle of a thin metal plate. 
Looking at a side view of a square-edged concentric orifice plate reveals sharp edges (90° corners) 
at the hole: 



Square-edged, concentric orifice plate 

(front view) 

(side view) 
Sharp edge 




Square-edged orifice plates may be installed in either direction, since the orifice plate "appears" 
exactly the same from either direction of fluid approach. In fact, this allows square-edged orifice 
plates to be used for measuring bidirectional flow rates (where the fluid flow direction reverses itself 
from time to time). A text label printed on the "paddle" of any orifice plate customarily identifies 
the upstream side of that plate, but in the case of the square-edged orifice plate it does not matter. 

The purpose of having a square edge on the hole in an orifice plate is to minimize contact with 
the fast-moving moving fluid stream going through the hole. Ideally, this edge will be knife-sharp. 
If the orifice plate is relatively thick (1/8 or an inch or more), it may be necessary to bevel the 
downstream side of the hole to further minimize contact with the fluid stream: 




Square-edged, concentric orifice plate 
with downstream bevel 

(front view) 

(side view) 


Looking at the side-view of this orifice plate, the intended direction of flow is left-to-right, with 
the sharp edge facing the incoming fluid stream and the bevel providing a non-contact outlet for 
the fluid. Beveled orifice plates are obviously uni-directional, and must be installed with the paddle 
text facing upstream. 

Other square-edged orifice plates exist to address conditions where gas bubbles or solid particles 
may be present in liquid flows, or where liquid droplets or solid particles may be present in gas flows. 
The first of this type is called the eccentric orifice plate, where the hole is located off-center to allow 
the undesired portions of the fluid to pass through the orifice rather than build up on the upstream 



Square-edged, eccentric orifice plate 

(front view) 

(side view) 




For gas flows, the hole should be offset downward, so that any liquid droplets or solid particles 
may easily pass through. For liquid flows, the hole should be offset upward to allow gas bubbles to 
pass through and offset downward to allow heavy solids to pass through. 

The second off-center orifice plate type is called the segmental orifice plate, where the hole is not 
circular but rather just a segment of a concentric circle: 



Square-edged, segmental orifice plate 

(front view) 

(side view) 


As with the eccentric orifice plate design, the segmental hole should be offset downward in gas 
flow applications and either upward or downward in liquid flow applications depending on the type 
of undesired material(s) in the flowstream. 

Some orifice plates employ non-square-edged holes for the purpose of improving performance at 
low Reynolds number 6 values, where the effects of fluid viscosity are more apparent. These orifice 
plate types employ rounded- or conical-entrance holes in an effort to minimize the effects of fluid 
viscosity. Experiments have shown that decreased Reynolds number causes the flowstream to not 
contract as much when traveling through an orifice, thus limiting fluid acceleration and decreasing the 
amount of differential pressure produced by the orifice plate. However, experiments have also shown 
that decreased Reynolds number in a venturi-type flow element causes an increase in differential 
pressure due to the effects of friction against the entrance cone walls. By manufacturing an orifice 
plate in such a way that the hole exhibits "venturi-like" properties (i.e. a dull edge where the fast- 
moving fluid stream has more contact with the plate), these two effects tend to cancel each other, 

3 To read more about the concept of Reynolds number, refer to section 1.8.9 beginning on page 51. 



resulting in an orifice plate that maintains consistent accuracy at lower flow rates and/or higher 
viscosities than the simple square-edged orifice. 

Two common non-square-edge orifice plate designs are the quadrant-edge and conic- entrance 
orifices. The quadrant-edge is shown first: 



Quadrant-edge orifice plate 

(front view) 




(side view) 

The conical-entrance orifice plate looks like a beveled square-edge orifice plate installed 
backwards, with flow entering the conical side and exiting the square-edged side: 



Conical-entrance orifice plate 

(front view) 




(side view) 

Here, is it vitally important to pay attention to the paddle's text label. This is the only sure 
indication of which direction an orifice plate needs to be installed. One can easily imagine an 
instrument technician mistaking a conical-entrance orifice plate for a square-edged, beveled orifice 
plate and installing it backward! 

Several standards exist for pressure tap locations. Ideally, the upstream pressure tap will detect 
fluid pressure at a point of minimum velocity, and the downstream tap will detect pressure at the 
vena contracta (maximum velocity). In reality, this ideal is never perfectly achieved. An overview 
of the most popular tap locations for orifice plates is shown in the following illustration: 



Flange taps 

Orifice plate 

Vena contracta taps 

Radius taps 

Corner taps 


Pipe taps or Full- flow taps 

Flange taps are the most popular tap location in the United States. Flanges may be manufactured 
with tap holes pre-drilled and finished before the flange is even welded to the pipe, making this a 
very convenient pressure tap configuration. Most of the other tap configurations require drilling into 
the pipe after installation, which is not only labor-intensive, but may possibly weaken the pipe at 
the locations of the tap holes. 

Corner taps must be used on small pipe diameters, where the vena contracta is so close to the 
downstream face of the orifice plate that a downstream flange tap would sense pressure in the highly 
turbulent region (too far downstream). Corner taps obviously require special (i.e. expensive) flange 
fittings, which is why they tend to be used only when necessary. 



Care should be taken to avoid measuring downstream pressure in the highly turbulent region 
following the vena contracta. This is why the pipe tap (also known as full-flow tap) standard calls 
for a downstream tap location eight pipe diameters away from the orifice: to give the flow stream 
room to stabilize for more consistent pressure readings 7 . 

Wherever the taps are located, it is vitally important that the tap holes be flush with the inside 
wall of the pipe or flange. Even the smallest recess or burr left from drilling will cause measurement 

For relatively low flow rates, an alternative arrangement is the integral orifice plate. This is 
where a small orifice plate is physically attached to the differential pressure-sensing element so that 
no impulse lines are needed. A photograph of an integral orifice plate and transmitter is shown here: 

7 What this means is that a "pipe tap" installation is actually measuring permanent pressure loss, which also 
happens to scale with the square of flow rate because the primary mechanism for energy loss in turbulent flow 
conditions is the translation of linear velocity to angular (swirling) velocity in the form of eddies. This kinetic energy 
is eventually dissipated in the form of heat as the eddies eventually succumb to viscosity. 



15.1.3 Other differential producers 

Other pressure-based flow elements exist as alternatives to the orifice plate. The Pitot tube, for 
example, senses pressure as the fluid stagnates (comes to a complete stop) against the open end of 
a forward-facing tube. A shortcoming of the classic single-tube Pitot assembly is sensitivity to fluid 
velocity at just one point in the pipe, so a more common form of Pitot tube seen in industry is the 
averaging Pitot tube consisting of several stagnation holes sensing velocity at multiple points across 
the width of the flow: 


Pitot tube 

pitot tube 

Pipe wall 


Pipe wall 



Pipe wall 

A variation on the latter theme is the Annubar flow element, a trade name of the Dieterich 
Standard corporation. An "Annubar" is an averaging pitot tube consolidating high and low pressure- 
sensing ports in a single probe assembly: 





Pipe wall 


\ Flow 


Pipe wall 


„ Holes 


Pipe wall 

A less sophisticated realization of the stagnation principle is the target flow sensor, consisting of 



a blunt "paddle" (or "drag disk") inserted into the flowstream. The force exerted on this paddle 
by the moving fluid is sensed by a special transmitter mechanism, which then outputs a signal 
corresponding to flow rate (proportional to the square of fluid velocity, just like an orifice plate): 


Pipe wall - - 




Pipe wall 

Drag disk <! 

Pipe wall 

The classic venturi tube pioneered by Clemens Herschel in 1887 has been adapted in a variety 
of forms broadly classified as flow tubes. All flow tubes work on the same principle: developing a 
differential pressure by channeling fluid flow from a wide tube to a narrow tube. The differ from the 
classic venturi only in construction details. Examples of flow tube designs include the Dall tube, 
Lo-Loss flow tube, Gentile or Bethlehem flow tube, and the B.I.F. Universal Venturi. 

Another variation on the venturi theme is called a flow nozzle, designed to be clamped between 
the faces of two pipe flanges in a manner similar to an orifice plate. The goal here is to achieve 
simplicity of installation approximating that of an orifice plate while improving performance (less 
permanent pressure loss) over orifice plates: 




ID H-*-V 2 D- 


Flow nozzle 

Two more variations on the venturi theme are the V-cone and Segmental wedge flow elements. 
The V-cone (or "venturi cone," a trade name of the McCrometer division of the Danaher corporation) 
may be thought of as a venturi tube in reverse: instead of narrowing the tube's diameter to cause 
fluid acceleration, fluid must flow around a cone-shaped obstruction placed in the middle of the tube. 
The tube's effective area will be reduced by the presence of this cone, causing fluid to accelerate 
through the restriction just as it would through the throat of a classic venturi tube: 




This cone is hollow, with a pressure-sensing port on the downstream side allowing for easy 
detection of fluid pressure near the vena contracta. Upstream pressure is sensed by another port in 



the pipe wall upstream of the cone. The following photograph shows a V-cone flow tube, cut away 
for demonstration purposes: 

Segmental wedge elements are special pipe sections with wedge-shaped restrictions built in. These 
devices, albeit crude, are useful for measuring the flow rates of slurries, especially when pressure is 
sensed by the transmitter through remote-seal diaphragms (to eliminate the possibility of impulse 
tube plugging): 

Segmental wedge 





Finally, the lowly pipe elbow may be pressed into service as a flow- measuring element, since fluid 
turning a corner in the elbow experiences radial acceleration and therefore generates a differential 
pressure along the axis of acceleration: 

Pipe elbow 

Pipe elbows should be considered for flow measurement only as a last resort. Their inaccuracies 
tend to be extreme, owing to the non-precise construction of most pipe elbows and the relatively 
weak differential pressures generated 8 . 

8 The fact that a pipe elbow generates small differential pressure is an accuracy concern because other sources of 
pressure become larger by comparison. Noise generated by fluid turbulence in the elbow, for example, becomes a 
significant portion of the pressure sensed by the transmitter when the differential pressure is so low (i.e. the signal-to- 
noise ratio becomes smaller). Errors caused by differences in elbow tap elevation and different impulse line fill fluids, 
for example, become more significant as well. 



15.1.4 Proper installation 

Perhaps the most common way in which the flow measurement accuracy of any flowmeter becomes 
compromised is incorrect installation, and pressure-based flowmeters are no exception to this rule. 
The following list shows some of the details one must consider in installing a pressure-based flowmeter 

• Necessary upstream and downstream straight-pipe lengths 

• Beta ratio 

• Impulse tube tap locations 

• Tap finish 

• Transmitter location in relation to the pipe 

Sharp turns in piping networks introduce large-scale turbulence into the flowstream. Elbows, 
tees, valves, fans, and pumps are some of the most common causes of large-scale turbulence in 
piping systems. When the natural flow path of a fluid is disturbed by such a device, the velocity 
profile of that fluid will become asymmetrical; e.g. the velocity gradient from one wall boundary 
of the pipe to the other will not be orderly. Large eddies in the flowstream (called swirl) will be 
present. This may cause problems for pressure-based flow elements which rely on linear acceleration 
(change in velocity in one dimension) to measure fluid flow rate. If the flow profile is distorted 
enough, the acceleration detected at the element may be too great or too little, and therefore not 
properly represent the full fluid flowstream 9 . 

Large-scale disturbances 

Even disturbances located downstream of the flow element impact measurement accuracy (albeit 
not as much as upstream disturbances). Unfortunately, both upstream and downstream flow 
disturbances are unavoidable on all but the simplest fluid systems. This means we must devise 
ways to stabilize a flowstream's velocity profile near the flow element in order to achieve accurate 
measurements of flow rate. A very simple and effective way to stabilize a flow profile is to provide 
adequate lengths of straight pipe ahead of (and behind) the flow element. Given enough time, even 
the most chaotic flowstream will "settle down" to a symmetrical profile all on its own. The following 

9 L.K. Spink mentions in his book Principles and Practice of Flow Meter Engineering that certain tests have shown 
flow measurement errors induced from severe disturbances as far as 60 to 100 pipe diameters upstream of the primary 
flow element! 



illustration shows the effect of a pipe elbow on a flowstream, and how the velocity profile returns to 
a normal (symmetrical) form after traveling through a sufficient length of straight pipe: 

Recommendations for minimum upstream and downstream straight-pipe lengths vary 
significantly with the nature of the turbulent disturbance, piping geometry, and flow element. As 
a general rule, elements having a smaller beta ratio (ratio of throat diameter d to pipe diameter 
D) are more tolerant of disturbances, with profiled flow (e.g. venturi tubes, flow tubes) having the 
greatest tolerance 10 . Ultimately, you should consult the flow element manufacturer's documentation 
for a more detailed recommendation appropriate to any specific application. 

In applications where sufficient straight-run pipe lengths are impractical, another option exists 
for "taming" turbulence generated by piping disturbances. Devices called flow conditioners may be 
installed upstream of the flow element to help the flow profile achieve symmetry in a far shorter 
distance than simple straight pipe could do alone. Flow conditioners take the form of a series of 
tubes or vanes installed inside the pipe, parallel to the direction of flow. These tubes or vanes force 
the fluid molecules to travel in straighter paths, thus stabilizing the flowstream prior to entering a 
flow element: 

Flow conditioner 

Velocity profile 

Another common source of trouble for pressure-based flowmeters is improper transmitter 
location. Here, the type of process fluid flow being measured dictates how the pressure-sensing 
instrument should be located in relation to the pipe. For gas and vapor flows, it is important that 
no stray liquid droplets collect in the impulse lines leading to the transmitter, lest a vertical liquid 
column begin to collect in those lines and generate an error-producing pressure. For liquid flows, it is 
important that no gas bubbles collect in the impulse lines, or else those bubbles may displace liquid 
from the lines and thereby cause unequal vertical liquid columns, which would (again) generate an 
error-producing differential pressure. 

In order to let gravity do the work of preventing these problems, we must locate the transmitter 
above the pipe for gas flow applications and below the pipe for liquid flow applications: 

10 However, there are disadvantages to using small-beta elements, one of them being increased permanent pressure 
loss which usually translates to increased operating costs due to energy loss. 




Proper mounting 
position for measuring 
gas flow 



Proper mounting 
position for measuring 
liquid flow 

Proper mounting 
position for measuring 
gas flow 



a ihti inii Co 

W w w w 



Proper mounting 
position for measuring 
liquid flow 

Condensible vapor applications (such as steam flow measurement) should be treated the same 
as liquid measurement applications. Here, condensed liquid will collect in the transmitter's impulse 
lines so long as the impulse lines are cooler than the vapor flowing through the pipe (which is 
typically the case). Placing the transmitter below the pipe allows vapors to condense and fill the 


impulse lines with liquid (condensate), which then acts as a natural seal protecting the transmitter 
from exposure to hot process vapors. 

In such applications it is important for the technician to pre- fill both impulse lines with condensed 
liquid prior to placing the flowmeter into service. "Tee" fittings with removable plugs or fill valves 
are provided to do this. Failure to pre-fill the impulse lines will likely result in measurement errors 
during initial operation, as condensed vapors will inevitably fill the impulse lines at slightly different 
rates and cause a difference in vertical liquid column heights within those lines. 

If tap holes must be drilled into the pipe (or flanges) at the process site, great care must be 
taken to properly drill and de-burr the holes. A pressure-sensing tap hole should be flush with the 
inner pipe wall, with no rough edges or burrs to create turbulence. Also, there should be no reliefs 
or countersinking near the hole on the inside of the pipe. Even small irregularities at the tap holes 
may generate surprisingly large flow-measurement errors. 


15.1.5 High-accuracy flow measurement 

Many assumptions were made in formulating flow equations from physical conservation laws. Suffice 
it to say, the flow formulae you have seen so far in this chapter are only approximations of reality. 
Orifice plates are some of the worst offenders in this regard, since the fluid encounters such abrupt 
changes in geometry passing through the orifice. Venturi tubes are nearly ideal, since the machined 
contours of the tube ensure gradual changes in fluid pressure and minimize turbulence. 

However, in the real world we must often do the best we can with imperfect technologies. Orifice 
plates, despite being less than perfect as flow-sensing elements, are convenient and economical to 
install in flanged pipes. Orifice plates are also the easiest type of flow element to replace in the event 
of damage or routine servicing. In applications such as custody transfer, where the flow of fluid 
represents product being bought and sold, flow measurement accuracy is paramount. It is therefore 
important to figure out how to coax the most accuracy from the common orifice plate in order that 
we may measure fluid flows both accurately and economically. 

If we compare the true flow rate through a pressure-generating primary sensing element 
against the theoretical flow rate predicted by an idealized equation, we may notice a substantial 
discrepancy 11 . Causes of this discrepancy include, but are not limited to: 

• Energy losses due to turbulence and viscosity 

• Energy losses due to friction against the pipe and element surfaces 

• Uneven flow profile, especially at low Reynolds numbers 

• Fluid compressibility 

• Thermal expansion (or contraction) of the element 

• Non-ideal pressure tap location(s) 

The ratio between true flow rate and theoretical flow rate for any measured amount of differential 
pressure is known as the discharge coefficient of the flow-sensing element, symbolized by the variable 
C. Since a value of 1 represents a theoretical ideal, the actual value of C for any real pressure- 
generating flow element will be less than 1: 

True flow 

Theoretical flow 

For gas and vapor flows, true flow rate deviates even more from the theoretical (ideal) flow value 
than liquids do, for reasons that have to do with the compressible nature of gases and vapors. A gas 
expansion factor (Y) may be calculated for any flow element by comparing its discharge coefficient 
for gases against its discharge coefficient for liquids. As with the discharge coefficient, values of Y 
for any real pressure-generating element will be less than 1: 

y ^gas 



11 Richard W. Miller, in his book Flow Measurement Engineering Handbook, states that venturi tubes may come 
within 1 to 3 percent of ideal, while a square-edged orifice plate may perform as poorly as only 60 percent of theoretical! 



( True gas flow 
V Theoretical gas flow 
I True liquid flow 
V Theoretical liquid flow 

Incorporating these factors into the ideal volumetric flow equation developed on page 452, 
arrive at the following formulation: 

If we wished, we could even add another factor to account for any necessary unit conversions 
(N), getting rid of the constant \f2 in the process: 

Q = N- CYA > l P i- P > 

! ,a s 


Sadly, neither the discharge coefficient (C) nor the gas expansion factor (Y) will remain constant 
across the entire measurement range of any given flow element. These variables are subject to some 
change with flow rate, which further complicates the task of accurately inferring flow rate from 
differential pressure measurement. However, if we know the values of C and Y for typical flow 
conditions, we may achieve good accuracy most of the time. 

Likewise, the fact that C and Y change with flow places limits on the accuracy obtainable with 
the "proportionality constant" formulae seen earlier. Whether we are measuring volumetric or mass 
flow rate, the k factor calculated at one particular flow condition will not hold constant for all flow 

Q = J Pl - P2 

W = ky/p(P! - P 2 ) 

This means after we have calculated a value for k based on a particular flow condition, we can 
only trust the results of the equation for flow conditions not too different from the one we used to 
calculate k. 

As you can see in both flow equations, the density of the fluid (p) is an important factor. If 
fluid density is relatively stable, we may treat p as a constant, incorporating its value into the 
proportionality factor (k) to make the two formulae even simpler: 

Q = k Q ^P 1 -P 2 

W = k wv / P 1 -P 2 

However, if fluid density is subject to change over time, we will need some means to continually 
calculate p so that our inferred flow measurement will remain accurate. Variable fluid density is a 


typical state of affairs in gas flow measurement, since all gases are compressible by definition. A 
simple change in static gas pressure within the pipe is all that is needed to make p change, which in 
turn affects the relationship between flow rate and differential pressure drop. 

The American Gas Association (AGA) provides a formula for calculating volumetric flow of any 
gas using orifice plates in their #3 Report, compensating for changes in gas pressure and temperature. 
A variation of that formula is shown here (consistent with previous forms in this section) : 

Q = N 

CYA 2 Z S P 1 (P 1 -P 2 

^ 2 V G f z n T 


Q = Volumetric flow rate (e.g. gallons per minute, standard cubic feet per second) 

N = Unit conversion factor 

C = Discharge coefficient (accounts for energy losses, Reynolds number corrections, pressure tap 
locations, etc.) 

A\ = Cross-sectional area of mouth 

A 2 = Cross-sectional area of throat 

Z s = Compressibility factor of gas under standard conditions 

Zfl = Compressibility factor of gas under flowing conditions, upstream 

Gf = Specific gravity of gas 

T = Absolute temperature of gas 

Pi = Upstream pressure (absolute) 

P 2 = Downstream pressure (absolute) 

This equation implies the continuous measurement of gas pressure (Pi) and temperature (T) 
inside the pipe, in addition to the differential pressure produced by the orifice plate (Pi — P 2 ). 
These measurements may be taken by three separate devices, their signals routed to a gas flow 




! ■ 

■---< — i 




Flow signal to 
-- flow indicator, 
flow controller, 



Orifice plate 

Note the location of the RTD (thermowell) , positioned downstream of the orifice plate so that 
the turbulence it generates will have negligible impact on the fluid dynamics at the orifice plate. 
The American Gas Association (AGA) allows for upstream placement of the thermowell, but only 
if located at least three feet upstream of a flow conditioner 12 . 

This photograph shows an AGA3-compliant installation of several orifice plates to measure the 
flow of natural gas: 

Specified in Part 2 of the AGA Report #3, section 2.6.5, page 22. 



~"F tflT^^^^j] 





■ 1 1 


IHH h5 

Llf r ^f. Z 

fffH ~~1 

Note the special transmitter manifolds, built to accept both the differential pressure and absolute 
pressure (Rosemount model 3051) transmitters. Also note the quick-change fittings (the ribbed 
cast-iron housings) holding the orifice plates (which cannot be directly seen) , to facilitate convenient 
change-out of the orifice plates which is periodically necessary due to wear. It is not unheard of 
to replace orifice plates on a daily basis to ensure the sharp orifice edges necessary for accurate 
measurement 13 . 

An alternative strategy is to use a single multi-variable transmitter capable of measuring gas 
temperature as well as both static and differential pressures. This approach enjoys the advantage of 
simpler installation over the multi-instrument approach: 

13 This is especially true in the gas exploration industry, where natural gas coming out of the earth is laden with a 
substantial amount of sand, rocks, and grit. 



Digital bus 
-o o- 

Multivariable transmitter 
(measures static pressure, 
differential pressure, and 
temperature in one unit) 

Orifice plate 

The Rosemount model 3095MV and Yokogawa model EJX910 are examples of multi-variable 
transmitters designed to perform compensated gas flow measurement, equipped with multiple 
pressure sensors, a connection port for an RTD temperature sensor, and sufficient digital computing 
power to continuously calculate flow rate based on the AGA equation. Such multi-variable 
transmitters may provide an analog output for computed flow rate, or a digital output where all 
three primary variables and the computed flow rate may be transmitted to a host system (as shown 
in the previous illustration). The Yokogawa EJX910A provides an interesting signal output option: 
a digital pulse signal, where each pulse represents a specific quantity (either volume or mass) of fluid. 
The frequency of this pulse train represents flow rate, while the total number of pulses counted over 
a period of time represents the total amount of fluid that has passed through the orifice plate over 
that amount of time. 

Liquid flow measurement applications may also benefit from compensation, because liquid 
density changes with temperature. Static pressure is not a concern here, because liquids are 
considered incompressible for all practical purposes 14 . Thus, the formula for compensated liquid 
flow measurement does not include any terms for static pressure, just differential pressure and 


CYA 2 



(P 1 - P 2 )[l + k T (T - T ref )] 

The constant kx shown in the above equation is the proportionality factor for liquid expansion 
with increasing temperature. The difference in temperature between the measured condition (T) 
and the reference condition (T re /) multiplied by this factor determines how much less dense the 
liquid is compared to its density at the reference temperature. 

14 Liquids can and do compress, the measurement of their "compressibility" being what is called the bulk modulus. 
However, this compressibility is too slight to be of any consequence in most flow measurement applications. 


15.1.6 Equation summary 

Volumetric flow rate (Q): 

Q = N- CYA > l P i- P > 



Mass flow rate (W): 


il =-V , = =, „r(Pi -P. 



Q = Volumetric flow rate (e.g. gallons per minute, flowing cubic feet per second) 

W = Mass flow rate (e.g. kilograms per second, slugs per minute) 

N = Unit conversion factor 

C = Discharge coefficient (accounts for energy losses, Reynolds number corrections, pressure tap 
locations, etc.) 

Y = Gas expansion factor (Y = 1 for liquids) 

A\ = Cross-sectional area of mouth 

A-2 = Cross-sectional area of throat 

Pf = Fluid density at flowing conditions (actual temperature and pressure at the element) 

The beta ratio (j3) of a differential-producing element is the ratio of throat diameter to mouth 
diameter (j3 = 4k). This is the primary factor determining acceleration as the fluid increases velocity 
entering the constricted throat of a flow element (venturi tube, orifice plate, wedge, etc.). The 
following expression is often called the velocity of approach factor (commonly symbolized as E v ), 
because it relates the velocity of the fluid through the constriction to the velocity of the fluid as it 
approaches the flow element: 

EL = — , = Velocity of approach factor 

This same velocity approach factor may be expressed in terms of mouth and throat areas (A\ 
and A 2 , respectively): 

E v = — ! = = Velocity of approach factor 

A L 

When computing the volumetric flow of a gas in standard volume units (e.g. SCFM), the 
equation becomes much more complex than the simple (flowing) volumetric rate equation. Any 
equation computing flow in standard units must predict the effective expansion of the gas if it were 
to transition from flowing conditions (the actual pressure and temperature it experiences flowing 
through the pipe) to standard conditions (one atmosphere pressure at 60 degrees Fahrenheit). The 


compensated gas flow measurement equation published by the American Gas Association (AGA 
Report #3) in 1992 for orifice plates with flange taps calculates this expansion to standard conditions 
with a series of factors accounting for flowing and standard ("base") conditions, in addition to the 
more common factors such as velocity of approach and gas expansion. Most of these factors are 
represented in the AGA3 equation by different variables beginning with the letter F: 

Q = F n (F c + F s i)YF p bF t bF t fF gr Fp V y'hwPfi 


Q = Volumetric flow rate (standard cubic feet per hour - SCFH) 

F n = Numeric conversion factor (accounts for certain numeric constants, unit-conversion 
coefficients, and the velocity of approach factor E v ) 

F c = Orifice calculation factor (a polynomial function of the orifice plate's j3 ratio and Reynolds 
number), appropriate for flange taps 

F s i = Slope factor (another polynomial function of the orifice plate's [3 ratio and Reynolds 
number), appropriate for flange taps 

F c + F s i = Cd = Discharge coefficient, appropriate for flange taps 

Y = Gas expansion factor (a function of /?, differential pressure, static pressure, and specific 

F p b = Base pressure factor = 7 p , with pressure in PSIA (absolute) 

F t b = Base temperature factor = fto 4 ^, with temperature in degrees Rankine 

F t f = Flowing temperature factor = . / jV , with temperature in degrees Rankine 

F gr = Real gas relative density factor = a/tj- 

F pv = Supercompressibility factor = . / ~^ k - 

hw = Differential pressure produced by orifice plate (inches water column) 
Pfi = Flowing pressure of gas at the upstream tap (PSI absolute) 



15.2 Laminar flowmeters 

A unique form of differential pressure-based flow measurement deserves its own section in this flow 
measurement chapter, and that is the laminar flowmeter. 

Laminar flow is a condition of fluid motion where viscous (internal fluid friction) forces greatly 
overshadow inertial (kinetic) forces. A flowstream in a state of laminar flow exhibits no turbulence, 
with each fluid molecule traveling in its own path, with limited mixing and collisions with adjacent 
molecules. The dominant mechanism for resistance to fluid motion in a laminar flow regime is 
friction with the pipe or tube walls. Laminar flow is qualitatively predicted by low values of Reynolds 

This pressure drop created by fluid friction in a laminar flowstream is quantifiable, and is 
expressed in the Hagen-Poiseuille equation: 

Q = k 

/APD 4 


Q = Flow rate 

AP = Pressure dropped across a length of pipe 

D = Pipe diameter 

[i = Fluid viscosity 

L = Pipe length 

k = Coefficient accounting for units of measurement 

Laminar flowmeter elements generally consist of one or more tubes whose length greatly exceeds 
the inside diameter, arranged in such a way as to produce a slow-moving flow velocity. An example 
is shown here: 


Laminar flowmeter 


The expanded diameter of the flow element ensures a lower fluid velocity than in the pipes 
entering and exiting the element. This decreases the Reynolds number to the point where the flow 


regime exhibits laminar behavior. The large number of small-diameter tubes packed in the wide area 
of the element provide adequate wall surface area for the fluid's viscosity to act upon, creating an 
overall pressure drop from inlet to outlet which is measured by the differential pressure transmitter. 
This pressure drop is permanent (no recovery of pressure downstream) because the mechanism of 
pressure drop is friction: total dissipation (loss) of energy in the form of heat. 

Another common form of laminar flow element is simply a coiled capillary tube: a long tube 
with a very small inside diameter. The small inside diameter of such a tube makes wall-boundary 
effects dominant, such that the flow regime will remain laminar over a wide range of flow rates. The 
extremely restrictive nature of a capillary tube, of course, limits the use of such flow elements to very 
low flow rates such as those encountered in the sampling networks of certain analytical instruments. 

A unique advantage of the laminar flowmeter is its linear relationship between flow rate and 
developed pressure drop. It is the only pressure-based flow measurement device for filled pipes 
that exhibits a linear pressure/flow relationship. This means no "square-root" characterization 
is necessary to obtain linear flow measurements with a laminar flowmeter. The big disadvantage 
of this meter type is its dependence on fluid viscosity, which in turn is strongly influenced by 
fluid temperature. Thus, all laminar flowmeters require temperature compensation in order to 
derive accurate measurements, and some even use temperature control systems to force the fluid's 
temperature to be constant as it moves through the element 15 . 

Laminar flow elements find their widest application inside pneumatic instruments, where a linear 
pressure/flow relationship is highly advantageous (behaving like a "resistor" for instrument air flow) 
and the viscosity of the fluid (instrument air) is relatively constant. Pneumatic controllers, for 
instance, use laminar restrictors as part of the derivative and integral calculation modules, the 
combination of "resistance" from the restrictor and "capacitance" from volume chambers forming a 
sort of pneumatic time-constant (t) network. 

15 This includes elaborate oil-bath systems where the laminar flow element is submerged in a temperature-controlled 
oil bath, the purpose of which is to hold temperature inside the laminar element constant despite sudden changes in 
the measured fluid's temperature. 


15.3 Variable-area flowmeters 


An Variable-area flowmeter is one where the fluid must pass through a restriction whose area 
increases with flow rate. The simplest example of a variable-area flowmeter is the rotameter, which 
uses a solid object (called a plummet or float) as a flow indicator, suspended in the midst of a 
tapered tube: 


Scale z 


Flow i 

Clear, tapered 
"* glass tube 

Plummet, or 



As fluid flows upward through the tube, a pressure differential develops across the plummet. This 
pressure differential, acting on the effective area of the plummet body, develops an upward force 
(F = ^). If this force exceeds the weight of the plummet, the plummet moves up. As the plummet 
moves further up in the tapered tube, the area between the plummet and the tube walls (through 
which the fluid must travel) grows larger. This increased flowing area allows the fluid to make it past 
the plummet without having to accelerate as much, thereby developing less pressure drop across the 
plummet's body. At some point, the flowing area reaches a point where the pressure-induced force 
on the plummet body exactly matches the weight of the plummet. This is the point in the tube 
where the plummet stops moving, indicating flow rate by it position relative to a scale mounted (or 
etched) on the outside of the tube. 

The following rotameter uses a spherical plummet, suspended in a flow tube machined from a 
solid block of clear plastic. An adjustable valve at the bottom of the rotameter provides a means 
for adjusting gas flow: 



The same basic flow equation used for pressure-based flow elements holds true for rotameters as 



Pi -Pi 

However, the difference in this application is that the value of the radicand is constant, since the 
pressure difference will remain constant 16 and the fluid density will likely remain constant as well. 
Thus, fc will change in proportion to Q. The only variable within fc relevant to plummet position is 
the flowing area between the plummet and the tube walls. 

Most rotameters are indicating devices only. They may be equipped to transmit flow information 
electronically by adding sensors to detect the plummet's position in the tube, but this is not common 

16 If we know that the plummet's weight will remain constant, its area will remain constant, and that the force 
generated by the pressure drop will always be in equilibrium with the plummet's weight for any steady flow rate, then 
the relationship F = -r dictates a constant pressure. Thus, we may classify the rotameter as a constant-pressure, 
variable-area flowmeter. This stands in contrast to devices such as orifice plates, which are variable-pressure, constant- 



Rotameters are very commonly used as purge flow indicators for pressure and level measurement 
systems requiring a constant flow of purge fluid (see pages 338 and 361 for examples). Such 
rotameters are usually equipped with hand-adjustable needle valves for manual regulation of purge 
fluid flow rate. 

A very different style of variable-area flowmeter is used extensively to measure flow rate through 
open channels, such as irrigation ditches. If an obstruction is placed within a channel, any liquid 
flowing through the channel must rise on the upstream side of the obstruction. By measuring this 
liquid level rise, it is possible to infer the rate of liquid flow past the obstruction. 

The first form of open-channel flowmeter is the weir, which is nothing more than a dam 
obstructing passage of liquid through the channel. Three styles of weir are shown in the following 
illustration; the rectangular, Cippoletti, and V-notch: 




A rectangular weir has a notch of simple rectangular shape, as the name implies. A Cippoletti 
weir is much like a rectangular weir, except that the vertical sides of the notch have a 4:1 slope 
(rise of 4, run of 1; approximately a 14 degree angle from vertical). A V-notch weir has a triangular 
notch, customarily measuring either 60 or 90 degrees. 

The following photograph shows water flowing through a Cippoletti weir: 

At a condition of zero flow through the channel, the liquid level will be at or below the crest 
(lowest point on the opening) of the weir. As liquid begins to flow through the channel, it must spill 
over the crest of the weir in order to get past the weir and continue downstream in the channel. In 
order for this to happen, the level of the liquid upstream of the weir must rise above the weir's crest 



height. This height of liquid upstream of the weir represents a hydrostatic pressure, much the same 
as liquid heights in piezometer tubes represent pressures in a liquid flowstream through an enclosed 
pipe (see page 59 for examples of this). The height of liquid above the crest of a weir is analogous 
to the pressure differential generated by an orifice plate. As liquid flow is increased even more, a 
greater pressure (head) will be generated upstream of the weir, forcing the liquid level to rise. This 
effectively increases the cross-sectional area of the weir's "throat" as a taller stream of liquid exits 
the notch of the weir 17 . 

Greater level upstream 
of the weir 

Zero flow 

Liquid spilling 
over weir crest 

Effective notch area 

Effective notch area 

Zero flow 

Some flow 

More flow 

This dependence of notch area on flow rate creates a very different relationship between flow rate 
and liquid height (measured above the crest) than the relationship between flow rate and differential 
pressure in an orifice plate: 

Q = 3.33{L-0.2H)H 


Rectangular weir 

Q = 3.367LH 15 Cippoletti weir 

17 Orifice plates are variable-pressure, constant-area flowmeters. Rotameters are constant-pressure, variable-area 
flowmeters. Weirs are variable-pressure, variable-area flowmeters. As one might expect, the mathematical functions 
describing each of these flowmeter types is unique! 



Q = 2.48 tan 




V-notch weir 


Q = Volumetric flow rate (cubic feet per second - CFS) 
L = Width of crest (feet) 
9 = V-notch angle (degrees) 
H = Head (feet) 

As you can see from a comparison of characteristic flow equations between these three types of 
weirs, the shape of the weir's notch has a dramatic effect on the mathematical relationship between 
flow rate and head (liquid level upstream of the weir, measured above the crest height). This implies 
that it is possible to create almost any characteristic equation we might like just by carefully shaping 
the weir's notch in some custom form. A good example of this is the so-called proportional or Sutro 

This weir design is not used very often, due to its inherently weak structure and tendency to clog 
with debris. 

A variation on the theme of a weir is another open-channel device called a flume. If weirs may 
be thought of as open-channel orifice plates, then flumes may be thought of as open-channel venturi 



Like weirs, flumes generate upstream liquid level height changes indicative of flow rate. One of 
the most common flume design is the Parshall flume, named after its inventor R.L. Parshall when 
it was developed in the year 1920. 

The following formulae relate head (upstream liquid height) to flow rate for free-flowing Parshall 
flumes 18 : 

Q = 0.992H 1547 3-inch wide throat Parshall flume 
Q = 2.06H 158 6-inch wide throat Parshall flume 
Q = 3.07H 153 9-inch wide throat Parshall flume 

Q = 4LH 


1-foot to 8-foot wide throat Parshall flume 

Q= (3.6875L + 2.5) H 


10-foot to 50-foot wide throat Parshall flume 


Q = Volumetric flow rate (cubic feet per second - CFS) 
L = Width of flume throat (feet) 
H = Head (feet) 

Flumes are generally less accurate than weirs, but they do enjoy the advantage of being inherently 
self-cleaning. If the liquid stream being measured is drainage- or waste-water, a substantial amount 
of solid debris may be present in the flow that could cause repeated clogging problems for weirs. In 
such applications, flumes are often the more practical flow element for the task (and more accurate 
over the long term as well, since even the finest weir will not register accuracy once it becomes fouled 
by debris). 

Once a weir or flume has been installed in an open channel to measure the flow of liquid, some 
method must be employed to sense upstream liquid level and translate this level measurement into 

18 It is also possible to operate a Parshall flume in fully submerged mode, where liquid level must be measured at 
both the upstream and throat sections of the flume. Correction factors must be applied to these equations if the flume 
is submerged. 



a flow measurement. Perhaps the most common technology for weir/flume level sensing is ultrasonic 
(see section 13.5.1, page number 390, for more information on how this technology works). Ultrasonic 
level sensors are completely non-contact, which means they cannot become fouled by the process 
liquid (or debris in the process liquid). However, they may be "fooled" by foam or debris floating 
on top of the liquid, as well as waves on the liquid surface. 

The following photograph shows a Parshall flume measuring effluent flow from a municipal sewage 
treatment plant, with an ultrasonic transducer mounted above the middle of the flume to detect 
water level flowing through: 

Once the liquid level is successfully measured, a computing device is used to translate that 
level measurement into a suitable flow measurement (and in some cases even integrate that flow 
measurement with respect to time to arrive at a value for total liquid volume passed through the 
element, in accordance with the calculus relationship V = J Q dt + C). 

A technique for providing a clean and "quiet" (still) liquid surface to measure the level of is 
called a stilling well. This is an open-top chamber connected to the weir/flume channel by a pipe, 
so that the liquid level in the stilling well matches the liquid level in the channel. The following 
illustration shows a stilling well connected to a weir/flume channel, with the direction of liquid flow 
in the channel being perpendicular to the page (i.e. either coming toward your eyes or going away 
from your eyes): 



Water in weir/flume 

To discourage plugging of the passageway connecting the stilling well to the channel, a small 
flow rate of clean water may be introduced into the well. This forms a constant purge flow into the 
channel, flushing out debris that might otherwise find its way into the connecting passageway to 
plug it up. Note how the purge water enters the stilling well through a submerged tube, so it does 
not cause splashing on the water's surface inside the well which could cause measurement problems 
for the ultrasonic sensor: 


Sediment flushed out of passageway 


15.4 Velocity-based flowmeters 

The Law of Continuity for fluids states that the product of mass density (p), cross-sectional pipe 
area (A) and average velocity (v) must remain constant through any continuous length of pipe: 


If the density of the fluid is not subject to change as it travels through the pipe (a very good 
assumption for liquids), we may simplify the Law of Continuity by eliminating the density terms 
from the equation: 

A 1 W[= A 2 V2 

The product of cross-sectional pipe area and average fluid velocity is the volumetric flow rate of 
the fluid through the pipe (Q = Av). This tells us that fluid velocity will be directly proportional 
to volumetric flow rate given a known cross-sectional area and a constant density for the fluid 
flowstream. Any device able to directly measure fluid velocity is therefore capable of inferring 
volumetric flow rate of fluid in a pipe. This is the basis for velocity-based flowmeter designs. 



15.4.1 Turbine flowmeters 

Turbine flowmeters use a free-spinning turbine wheel to measure fluid velocity, much like a miniature 
windmill installed in the flow stream. The fundamental design goal of a turbine flowmeter is to make 
the turbine element as free-spinning as possible, so that no torque is required to sustain the turbine's 
rotation. If this goal is achieved, the turbine blades will achieve a rotating (tip) speed that equalizes 
with the linear velocity of the fluid: 

Turbine blades 

Fluid flow 

Turbine wheel 


Turbine shaft 

Direction of 
wheel rotation 

A cut-away demonstration model of a turbine flowmeter is shown in the following photograph. 
The blade sensor may be seen protruding from the top of the flowtube, just above the turbine wheel: 



Note the sets of "flow conditioner" vanes immediately before and after the turbine wheel in the 
photograph. As one might expect, turbine flowmeters are very sensitive to swirl in the process fluid 
flowstream. In order to achieve high accuracy, the flow profile must not be swirling in the vicinity 
of the turbine, lest the turbine wheel spin faster or slower than it should to represent the velocity 
of a straight-flowing fluid. 

Each blade on the turbine acts as an inclined plane for the fluid molecules as they pass by. The 
angle of the blades determines the ratio of tip speed to fluid velocity 19 . 

Turbine speed may be transmitted to an indicator mechanically by means of cables and/or gears, 
electronically by means of magnetic sensor using a "pickup" coil to generating voltage pulses as the 
turbine blades rotate underneath, or even optically in some applications by reflecting light off the 
specific locations on the turbine wheel (the light pulses conveyed to and from the wheel via fiber- 
optic cables). Pickup coils are preferred over mechanical cables or gears for the simple reason of 
less resistance to rotation. Cables and gears always present some degree of friction to the turbine's 
rotation, causing the flowmeter to register less flow than there actually is. Magnetic pickup sensors, 
however, are frictionless. 

The rotational speed of the turbine wheel directly relates to fluid velocity, which is proportional 
to volumetric flow rate. If a magnetic pickup is used to detect the turbine blades as they pass by the 
sensor, the frequency of the AC voltage signal output by the sensor relates directly to fluid velocity 
(and volumetric flow). 

19 For instance, a blade angle of 45 degrees would make blade tip speed equal to fluid velocity. A blade angle of only 
30 degrees (from the turbine shaft centerline) would result in a blade tip speed of about one-half fluid velocity. 


Since volumetric flow and pickup coil output frequency are directly proportional to each other, 
we may express this relationship in the form of an equation: 

f = kQ 


/ = Frequency of output signal (Hz) 

Q = Volumetric flow rate (e.g. gallons per second) 

k = "K" factor of the turbine element (e.g. pulses per gallon) 

Dimensional analysis confirms the validity of this equation. Using units of GPS (gallons per 
second) and pulses per gallon, we see that the product of these two quantities is indeed pulses per 
second (equivalent to cycles per second, or Hz): 



. S al . 




Using algebra to solve for flow (Q), we see that it is the quotient of frequency and K factor that 
yields a volumetric flow rate for a turbine flowmeter: 

If pickup signal frequency directly represents volumetric flow rate, then the total number of 
pulses accumulated in any given time span will represent the amount of fluid volume passed through 
the turbine meter over that same time span. We may express this algebraically as the product of 
average flow rate (Q), average frequency (/), K factor, and time: 


A more sophisticated way of calculating total volume passed through a turbine meter requires 
calculus, representing total volume as the time-integral of instantaneous signal frequency and K 
factor over a period of time from t = to t = T: 

V= J Qdt or V= J *-dt 

Jo Jo k 

We may achieve approximately the same result simply by using a digital counter circuit to 
totalize pulses output by the pickup coil and a microprocessor to calculate volume in whatever unit 
of measurement we deem appropriate. 

As with the orifice plate flow element, standards have been drafted for the use of turbine 
flowmeters as precision measuring instruments in gas flow applications, particularly the custody 
transfer 20 of natural gas. The American Gas Association has published a standard called the Report 
#7 specifying the installation of turbine flowmeters for high-accuracy gas flow measurement, along 

20 "Custody transfer" refers to measurement applications where a product is exchanging ownership. In other words, 
someone is selling, and someone else is buying, quantities of fluid as part of a business transaction. It is not difficult to 
understand why accuracy is important in such applications, as both parties have a vested interest in a fair exchange. 



with the associated mathematics for precisely calculating flow rate based on turbine speed, gas 
pressure, and gas temperature. 

The following photograph shows three AGA7-compliant installations of turbine flowmeters for 
measuring the flow rate of natural gas: 

Note the pressure-sensing and temperature-sensing instrumentation installed in the pipe, 
reporting gas pressure and gas temperature to a flow-calculating computer (along with turbine 
pulse frequency) for the calculation of natural gas flow rate. Less-critical applications may use a 
"compensated" turbine flowmeter that mechanically performs the same pressure- and temperature- 
compensation functions on turbine speed to achieve true gas flow measurement, as shown in the 
following photograph: 



The particular flowmeter shown in the above photograph uses a filled-bulb temperature sensor 
(note the coiled, armored capillary tube connecting the flowmeter to the bulb) and shows total gas 
flow by a series of pointers, rather than gas flow rate. 



15.4.2 Vortex flowmeters 

When a fluid moves with high Reynolds number past a stationary object (a "bluff body"), there is 
a tendency for the fluid to form vortices on either side of the object. Each vortex will form, then 
detach from the object and continue to move with the flowing gas or liquid, one side at a time 
in alternating fashion. This phenomenon is known as vortex shedding, and the pattern of moving 
vortices carried downstream of the stationary object is known as a vortex street. 

It is commonplace to see the effects of vortex shedding on a windy day by observing the motion 
of flagpoles, light poles, and tall smokestacks. Each of these objects has a tendency to oscillate 
perpendicular to the direction of the wind, owing to the pressure variations caused by the vortices 
as they alternately form and break away from the object: 


Flagpole (looking down from above) 

Side-to-side motion 
of the flagpole 

This alternating series of vortices was studied by Vincenc Strouhal in the late nineteenth century 
and later by Theodore von Karman in the early twentieth century. It was determined that the 
distance between successive vortices downstream of the stationary object is relatively constant, and 
directly proportional to the width of the object, for a wide range of Reynolds number values. If we 
view these vortices as crests of a continuous wave, the distance between vortices may be represented 
by the symbol customarily reserved for wavelength: the Greek letter "lambda" (A). 

Fluid flow 




1 — 1^ 



=t h— ^ 


The proportionality between object width (d) and vortex street wavelength (A) is called the 
Strouhal number (S), approximately equal to 0.17: 

XS = d 



If a differential pressure sensor is installed immediately downstream of the stationary object in 
such an orientation that it detects the passing vortices as pressure variations, an alternating signal 
will be detected: 



Fluid flow 





Pressure sensor 

The frequency of this alternating pressure signal is directly proportional to fluid velocity past 
the object, since the wavelength is constant. This follows the classic frequency- velocity- wavelength 
formula common to all traveling waves (A/ = v). Since we know the wavelength will be equal to 
the bluff body's width divided by the Strouhal number (approximately 0.17), we may substitute 
this into the frequency-velocity-wavelength formula to solve for fluid velocity (v) in terms of signal 
frequency (/) and bluff body width (d). 

v = Xf 



v = 


Thus, a stationary object and pressure sensor installed in the middle of a pipe section constitute 
a form of flowmeter called a vortex flowmeter. Like a turbine flowmeter with an electronic "pickup" 
sensor to detect the passage of rotating turbine blades, the output frequency of a vortex flowmeter 
is linearly proportional to volumetric flow rate. 

The pressure sensors used in vortex flowmeters are not standard differential pressure transmitters, 
since the vortex frequency is too high to be successfully detected by such bulky instruments. Instead, 
the sensors are typically piezoelectric crystals. These pressure sensors need not be calibrated, since 
the amplitude of the pressure waves detected is irrelevant. Only the frequency of the waves matter 
for measuring flow rate, and so nearly any pressure sensor with a fast enough response time will 



Like turbine meters, the relationship between sensor frequency (/) and volumetric flow rate 
(Q) may be expressed as a proportionality, with the letter k used to represent the constant of 
proportionality for any particular flowmeter: 

f = kQ 


/ = Frequency of output signal (Hz) 

Q = Volumetric flow rate (e.g. gallons per second) 

k = "K" factor of the vortex shedding flowtube (e.g. pulses per gallon) 

This means that vortex flowmeters, like electronic turbine meters, each have a particular "K 
factor" relating the number of pulses generated per unit volume passed through the meter 21 . 
Counting the total number of pulses over a certain time span yields total fluid volume passed 
through the meter over that same time span, making the vortex flowmeter readily adaptable for 
"totalizing" fluid volume just like turbine meters. 

Since vortex flowmeters have no moving parts, they do not suffer the problems of wear and 
lubrication facing turbine meters. There is no moving element to "coast" as in a turbine flowmeter 
if fluid flow suddenly stops, which means vortex flowmeters are better suited to measuring erratic 

The following photograph shows a vortex flow transmitter manufactured by Rosemount: 

The next two photographs show close-up views of the flowtube assembly, front (left) and rear 

21 This K factor is empirically determined for each flowmeter by the manufacturer using water as the test fluid (a 
factory "wet-calibration"), to ensure optimum accuracy. 





15.4.3 Magnetic flowmeters 

When an electrical conductor moves perpendicular to a magnetic field, a voltage is induced in 
that conductor perpendicular to both the magnetic flux lines and the direction of motion. This 
phenomenon is known as electromagnetic induction, and it is the basic principle upon which all 
electro-mechanical generators operate. 

In a generator mechanism, the conductor in question is typically a coil (or set of coils) made 
of copper wire. However, there is no reason the conductor must be made of copper wire. Any 
electrically conductive substance in motion is sufficient to electromagnetically induce a voltage, even 
if that substance is a liquid (or a gas 22 ). 

Consider water flowing through a pipe, with a magnetic field passing perpendicularly through 
the pipe: 


The direction of liquid flow cuts perpendicularly through the lines of magnetic flux, generating a 
voltage along an axis perpendicular to both. Metal electrodes opposite each other in the pipe wall 
intercept this voltage, making it readable to an electronic circuit. 

A voltage induced by the linear motion of a conductor through a magnetic field is called 
motional EMF, the magnitude of which is predicted by the following formula (assuming perfect 
perpendicularity between the direction of velocity, the orientation of the magnetic flux lines, and 
the axis of voltage measurement): 

2 Technically, a gas must be super- heated into a plasma state before it is able to conduct electricity. 


£ = Blv 


£ = Motional EMF (volts) 

B = Magnetic flux density (Tesla) 

I = Length of conductor passing through the magnetic field (meters) 

v = Velocity of conductor (meters per second) 

Assuming a fixed magnetic field strength (constant B) and an electrode spacing equal to the 
fixed diameter of the pipe (constant I = d), the only variable capable of influencing the magnitude 
of induced voltage is velocity (v). In our example, v is not the velocity of a wire segment, but rather 
the average velocity of the liquid flowstream (v) . Since we see that this voltage will be proportional 
to average fluid velocity, it must also be proportional to volumetric flow rate, since volumetric flow 
rate is also proportional to average fluid velocity 23 . Thus, what we have here is a type of flowmeter 
based on electromagnetic induction. These flowmeters are commonly known as magnetic flowmeters 
or simply mag-flow meters. 

We may state the relationship between volumetric flow rate (Q) and motional EMF {£) more 
precisely by algebraic substitution. First, we will write the formula relating volumetric flow to 
average velocity, and then manipulate it to solve for average velocity: 

Q = Av 


Next, we re-state the motional EMF equation, and then substitute ^ for v to arrive at an equation 
relating motional EMF to volumetric flow rate (Q), magnetic flux density (B) 7 pipe diameter (d), 
and pipe area (A): 

£ = Bdv 

£ = Bd^- 

t_ A 
Since we know this is a circular pipe, we know that area and diameter are directly related to 
each other by the formula A = ^— . Thus, we may substitute this definition for area into the last 
equation, to arrive at a formula with one less variable (only d, instead of both d and A): 



23 This is an application of the transitive property in mathematics: if two quantities are both equal to a common 
third quantity, they must also be equal to each other. This property applies to proportionalities as well as equalities: 
if two quantities are proportional to a common third quantity, they must also be proportional to each other. 


BdQ 4 


1 ird 2 


If we wish to have a formula defining flow rate Q in terms of motional EMF {£), we may simply 
manipulate the last equation to solve for Q: 

* AB 

This formula will successfully predict flow rate only for absolutely perfect circumstances. In order 
to compensate for inevitable imperfections, a "proportionality constant" (fc) is usually included in 
the formula 24 : 

O-k — 
Q ~ k AB 

Note the linearity of this equation. Nowhere do we encounter a power, root, or other non- 
linear mathematical function in the equation for a magnetic flowmeter. This means no special 
characterization is required to calculate volumetric flow rate. 

A few conditions must be met for this formula to successfully infer volumetric flow rate from 
induced voltage: 

• The liquid must be a reasonably good conductor of electricity 

• Both electrodes must contact the liquid 

• The pipe must be completely filled with liquid 

• The flowtube must be properly grounded to avoid errors caused by stray electric currents in 
the liquid 

The first condition is met by careful consideration of the process liquid prior to installation. 
Magnetic flowmeter manufacturers will specify the minimum conductivity value of the liquid to be 
measured. The second and third conditions are met by correct installation of the magnetic flowtube 
in the pipe. The installation must be done in such a way as to guarantee full flooding of the flowtube 
(no gas pockets). The flowtube is usually installed with electrodes across from each other horizontally 
(never vertically!) so that even a momentary gas bubble will not break electrical contact between 
an electrode tip and the liquid flowstream. 

Electrical conductivity of the process liquid must meet a certain minimum value, but that is 
all. It is surprising to some technicians that changes in liquid conductivity have little to no effect 
on flow measurement accuracy. It is not as though a doubling of liquid conductivity will result 
in a doubling of induced voltage! Motional EMF is strictly a function of physical dimensions, 
magnetic field strength, and fluid velocity. Liquids with poor conductivity simply present a greater 
electrical resistance in the voltage- measuring circuit, but this is of little consequence because the 
input impedance of the detection circuitry is phenomenally high. Common fluid types that will 

4 The colloquial term in the United States for this sort of thing is fudge factor. 



not work with magnetic flowmeters include deionized water (e.g. steam boiler feedwater, ultrapure 
water for pharmaceutical and semiconductor manufacturing) and oils. 

Proper grounding of the flowtube is very important for magnetic flowmeters. The motional 
EMF generated by most liquid flowstreams is very weak (1 millivolt or less!), and therefore may 
be easily overshadowed by noise voltage present as a result of stray electric currents in the piping 
and/or liquid. To combat this problem, magnetic flowmeters are usually equipped to shunt stray 
electric currents around the flowtube so that the only voltage intercepted by the electrodes will be 
the motional EMF produced by liquid flow. The following photograph shows a Rosemount model 
8700 magnetic flowtube, with braided-wire grounding straps clearly visible: 

Note how both grounding straps attach to a common junction point on the flowtube housing. 
This common junction point should also be bonded to a functional earth ground when the flowtube 
is installed in the process line. On this particular flowtube you can see a stainless steel grounding 
ring on the face of the near flange, connected to one of the braided grounding straps. An identical 
grounding ring lays on the other flange, but it is not clearly visible in this photograph. These rings 
provide points of electrical contact with the liquid in installations where the pipe is made of plastic, 



or where the pipe is metal but lined with a plastic material for corrosion resistance. 

Magnetic flowmeters are fairly tolerant of swirl and other large-scale turbulent fluid behavior. 
They do not require the long straight-runs of pipe upstream and downstream that orifice plates do, 
which is a great advantage in many piping systems. 

Some magnetic flowmeters have their signal conditioning electronics located integral to the 
flowtube assembly. A couple of examples are shown here (a pair of small Endress+Hauser flowmeters 
on the left and a large Toshiba flowmeter on the right): 

Other magnetic flowmeters have separate electronics and flowtube assemblies, connected together 
by shielded cable. In these installations, the electronics assembly is referred to as the flow transmitter 
(FT) and the flowtube as the flow element (FE): 



While in theory a permanent magnet should be able to provide the necessary magnetic flux for 
a magnetic flowmeter to function, this is almost never done in practice. The reason for this has to 
do with a phenomenon called polarization which occurs when a DC voltage is impressed across a 
liquid containing ions (electrically charged molecules). Ionic polarization would soon interfere with 
detection of the motional EMF if a magnetic flowmeter were to use a constant magnetic flux such 
as that produced by a permanent magnet. A simple solution to this problem is to alternate the 
polarity of the magnetic field, so that the motional EMF polarity also alternates and never gives the 
fluid ions enough time to polarize. 

This is why magnetic flowmeter tubes almost always use electromagnet coils to generate the 
magnetic flux necessary for induction to occur. A photograph of a Foxboro magnetic flowtube with 
one of the protective covers removed shows these wire coils clearly (in blue) : 

Perhaps the simplest form of coil excitation is when the coil is energized by 60 Hz AC power 
taken from the line power source. Since motional EMF is proportional to fluid velocity and to the 
flux density of the magnetic field, the induced voltage for such a coil will be a sine wave whose 
amplitude varies with volumetric flow rate. 

Unfortunately, if there is any stray electric current traveling through the liquid to produce 
erroneous voltage drops between the electrodes, chances are it will be 60 Hz AC as well. With 
the coil energized by 60 Hz AC, any such noise voltage may be falsely interpreted as fluid flow 
because the sensor electronics has no way to distinguish between 60 Hz noise in the fluid and a 60 
Hz motional EMF caused by fluid flow. 

A more sophisticated solution to this problem uses a pulsed excitation power source for the 
flowtube coils. This is called DC excitation by magnetic flowmeter manufacturers, which is a bit 


misleading because these "DC" excitation signals often reverse polarity, appearing more like an AC 
square wave on an oscilloscope display. The motional EMF for one of these flowmeters will bear the 
same waveshape, with amplitude once again being the indicator of volumetric flow rate. The sensor 
electronics can more easily reject any AC noise voltage because the frequency and waveshape of the 
noise (60 Hz, sinusoidal) will not match that of the flow-induced motional EMF signal. 


15.4.4 Ultrasonic flowmeters 

Ultrasonic flowmeters measure fluid velocity by passing high-frequency sound waves along the fluid 
flow path. Fluid motion influences the propagation of these sound waves, which may then be 
measured to infer fluid velocity. Two major sub-types of ultrasonic flowmeters exist: Doppler and 
transit-time. Both types of ultrasonic flowmeter work by transmitting a high-frequency sound wave 
into the fluid stream (the incident pulse) and analyzing the received pulse. 

Doppler flowmeters exploit the Doppler effect, which is the shifting of frequency resulting from 
waves emitted by or reflected by a moving object. Doppler flowmeters bounce sound waves off 
of bubbles or particulate material in the flow stream, measure the frequency shift, and infer fluid 

Fluid flow 



o ° 

o ° O 

If the reflected wave returns from a bubble that is advancing toward the flowmeter sensor, the 
reflected frequency will be greater than the incident frequency. If the flow reverses direction and the 
reflected wave returns from a bubble that is traveling away from the sensor, the reflected frequency 
will be less than the incident frequency. 

Doppler-effect ultrasonic flowmeters obviously require flowstream containing bubbles or 
particulate matter. In many applications this is a normal state of affairs (municipal wastewater, 
for example). However, some process fluids are simply too clean and too homogeneous to reflect 
sound waves. In such applications, a different sort of ultrasonic velocity detection technique must 
be applied. 

Transit-time flowmeters, sometimes called counterpropagation flowmeters, use a pair of opposed 
sensors to measure the time difference between a sound pulse traveling with the fluid flow versus a 
sound pulse traveling against the fluid flow. Since the motion of fluid tends to carry a sound wave 
along, the sound pulse transmitted downstream will make the journey faster than a sound pulse 
transmitted upstream: 



Fluid flow 


In this flowmeter design, a clean fluid with no solid impurities is essential for good signal 

One potential problem with the transit-time flowmeter is being able to measure the true average 
fluid velocity when the flow profile changes with Reynolds number. If just one ultrasonic "beam" is 
used to probe the fluid velocity, the path this beam takes will likely see a different velocity profile 
as the flow rate changes (and the Reynolds number changes along with it). Recall the difference 
in fluid velocity profiles between low Reynolds number flows (left) and high Reynolds number flows 

Laminar flow 

Turbulent flow 

Fluid flow 

•^ Velocity 

Fluid flow ^P^i$%=P ) Y^y 

A popular way to mitigate this problem is to use multiple sensor pairs, sending acoustic signals 
along multiple paths through the fluid (i.e. a multipath ultrasonic flowmeter), and to average the 
resulting velocity measurements. Dual-beam flowmeters have been in use for well over a decade, and 
one manufacturer even has a five beam ultrasonic flowmeter model which they claim maintains an 
accuracy of +/- 0.15% through the laminar-to-turbulent flow regime transition 25 . 

Some modern ultrasonic flowmeters have the ability to switch back and forth between Doppler 
and transit-time (counterpropagation) modes, automatically adapting to the fluid being sensed. This 
capability enhances the suitability of ultrasonic flowmeters to a wider range of process applications. 

Ultrasonic flowmeters are adversely affected by swirl and other large-scale fluid disturbances, 
and as such may require substantial lengths of straight pipe upstream and downstream of the 
measurement flowtube to stabilize the flow profile. 

Advances in ultrasonic flow measurement technology have reached a point where it is now feasible 
to consider ultrasonic flowmeters for custody transfer measurement of natural gas. The American 
Gas Association has released a report specifying the use of multipath ultrasonic flowmeters in this 
capacity (Report #9). 

25 See page 10 of Friedrich Hofmann's Fundamentals of Ultrasonic Flow Measurement for industrial applications 


A unique advantage to ultrasonic flow measurement is the ability to measure flow through 
the use of temporary clamp-on sensors rather than a specialized flowtube with built-in ultrasonic 
transducers. While clamp-on sensors are not without their share of problems 26 , they constitute an 
excellent solution for certain flow measurement applications. 

15.5 Inertia-based (true mass) flowmeters 

Flowmeters based on true mass measurement ignore fluid density, outputting a signal directly (and 
linearly) proportional to mass flow rate. These are quite useful in the chemical industries, where 
stoichiometric ratios must be accurately maintained. 

26 Most notably, the problem of achieving good acoustic coupling with the pipe wall so that signal transmission to 
the fluid and signal reception back to the sensor may be optimized. 



15.5.1 Coriolis flowmeters 

In physics, certain types of forces are classified as fictitious or pseudoforces because they only appear 
to exist when viewed from an accelerating perspective (called a non-inertial reference frame). The 
feeling you get in your stomach when you accelerate either up or down in an elevator, or when riding 
a roller-coaster at an amusement park, feels like a force acting against your body when it is really 
nothing more than the reaction of your body's inertia to being accelerated by the vehicle you are 
in. The real force is the force of the vehicle against your body, causing it to accelerate. What you 
perceive is merely a reaction to that force, and not the primary cause of your discomfort as it might 
appear to be. 

Centrifugal force is another example of a "pseudoforce" because although it may appear to be 
a real force acting on any rotating object, it is in fact nothing more than an inertial reaction. 
Centrifugal force is a common experience to any child who has ever played on a "merry-go-round:" 
that perception of a force drawing you away from the center of rotation, toward the rim. The real 
force acting on any rotating object is toward the center of rotation (a centripetal force) which is 
necessary to make the object radially accelerate toward a center point rather than travel in a straight 
line as it normally would without any forces acting upon it. When viewed from the perspective of 
the spinning object, however, it would seem there is a force drawing the object away from the center 
(a centrifugal force). 

Yet another example of a "pseudoforce" is the Coriolis force, more complicated than centrifugal 
force, arising from motion perpendicular to the axis of rotation in a non-inertial reference frame. 
The example of a merry-go-round works to illustrate Coriolis force as well: imagine sitting at the 
center of a spinning merry-go-round, holding a ball. If you gently toss the ball away from you and 
watch the trajectory of the ball, you will notice it curve rather than travel away in a straight line. In 
reality, the ball is traveling in a straight line (as viewed from an observer standing on the ground), 
but from your perspective on the merry-go-round, it appears to be deflected by an invisible force 
which we call the Coriolis force. 

In order to generate a Coriolis force, we must have a mass moving at a velocity perpendicular to 
an axis of rotation: 

Axis of rotation 

Apparent trajectory 

of the ball (as viewed 

from the rotating platform) 



The magnitude of this force is predicted by the following vector equation : 

F c = -2Q x v'm 


F c = Coriolis force vector 

il = Angular velocity (rotation) vector 

v' = Velocity vector as viewed from the rotating reference frame 

m = Mass of the object 

If we replace the ball with a fluid moving through a tube, and we introduce a rotation vector 
by tilting that tube around a stationary axis (a fulcrum), a Coriolis force develops on the tube in 
such a way as to oppose the direction of rotation just like the Coriolis force opposed the direction 
of rotation of the rotating platform in the previous illustration: 

Fluid motion 

Coriolis force 
.1 vector 

Axis of rotation 

To phrase this in anthropomorphic terms, the fluid "fights" against this rotation because it 
"wants" to keep traveling in a straight line. For any given rotational velocity, the amount of "fight" 
will be directly proportional to the product of fluid velocity and fluid mass. In other words, the 
magnitude of the Coriolis force will be in direct proportion to the fluid's mass flow rate. This is the 
basis of a Coriolis mass flowmeter. 

As you might guess, it can be difficult to engineer a tubing system capable of spinning in circles 
while carrying a flowstream of pressurized fluid. To bypass the practical difficulties of building 
a spinning tube system, Coriolis flowmeters are instead built on the principle of a flexible tube 
that oscillates back and forth, producing the same effect in an intermittent fashion rather than 
continuously. The effect is not unlike wiggling a hose side to side as it carries a stream of water: 

27 This is an example of a vector cross-product where all three vectors are perpendicular to each other, and the 
directions follow the right-hand rule. 




Arc of rotation 

(This illustration is from a vertical view, 
looking down. The Coriolis force acts 
laterally, bending the hose to the side.) 

We cannot build a Coriolis flowmeter exactly like the water hose illustration shown above unless 
we are willing to let the process fluid exit the tubing, so a common Coriolis flowmeter design uses 
a U-shaped tube that redirects the fluid flow back to the center of rotation. The curved end of the 
flexible U-tube is forced to shake back and forth while the tube ends anchor to a stationary manifold: 


The two parallel tubes will experience opposite Coriolis forces as the U-tube assembly shakes up 
and down, causing the U-bend to twist. As mass flow rate through the tube increases, so does the 
degree of twisting. By monitoring the amplitude of this twisting motion, we may infer the mass flow 
rate of the fluid passing through the tube: 





End view 

End view 

In order to reduce the amount of vibration generated by a Coriolis flowmeter, and more 
importantly to reduce the effect any external vibrations may have on the flowmeter, two identical U- 
tubes are built next to each other and shaken in complementary fashion (always moving in opposite 
directions) 28 . Tube twist is measured as relative motion from one tube to the next, not as motion 
between the tube and the stationary housing of the flowmeter. This (ideally) eliminates the effect 
of any common-mode vibrations on the inferred flow measurement: 

28 For those readers with an automotive bent, this is the same principle applied in opposed-cylinder engines (e.g. 
Porsche "boxer" air-cooled 6-cylinder engine, Volkswagen air-cooled 4-cylinder engine, BMW air-cooled motorcycle 
twin engine, Citroen 2CV 2-cylinder engine, Subaru 4- and 6-cylinder opposed engines, etc.). Opposite piston pairs are 
always 180° out of phase for the purpose of maintaining mechanical balance: both moving away from the crankshaft 
or both moving toward the crankshaft, at any given time. 



Great care is taken by the manufacturer to ensure the two tubes are as close to identical as 
possible: not only are their physical characteristics precisely matched, but the fluid flow is split very 
evenly between the tubes 29 so their respective Coriolis forces should be identical in magnitude. 

A photograph of a Rosemount (Micro-Motion) U-tube Coriolis flowmeter demonstration unit 
shows the U-shaped tubes (one tube is directly above the other in this picture, so you cannot tell 
there are actually two U-tubes): 

A closer inspection of this flowmeter shows that there are actually two U-tubes, one positioned 
directly above the other, shaken in complementary directions by a common electromagnetic force 

29 An alternative to splitting the flow is to plumb the tubes in series so they must share the exact same flow rate, 
like series-connected resistors sharing the exact same amount of electrical current. 



Two magnetic displacement sensors monitor the relative motions of the tubes and transmit 
signals to an electronics module for digital processing. One of those sensor coils may be seen in 
the previous photograph. Both the force coil and the sensor coil are nothing more than permanent 
magnets surrounded by movable copper wire coils. The main difference between the force coil and 
the sensor coil is that the force coil is powered by an AC signal to impart a vibratory force to the 
tubes, whereas the sensor coils are both unpowered so they can detect tube motion by generating AC 
voltages to be sensed by the electronics module. The force coil is shown in the left-hand photograph, 
while one of the two sensor coils appears in the right-hand photograph: 

Advances in sensor technology and signal processing have allowed the construction of Coriolis 
flowmeters employing straighter tubes than the U-tube unit previously illustrated and photographed. 
Straighter tubes are advantageous for reasons of reduced plugging potential and the ability to easily 
drain all liquids out of the flowmeter when needed. 



The tubes of a Coriolis flowmeter are not just conduits for fluid flow, they are also precision spring 
elements. As such, it is important to precisely know the spring constant value of these tubes so that 
the Coriolis force may be inferred from tube displacement (i.e. how far the tubes twist). Every 
Coriolis flow element is factory-tested to determine the flow tubes' mechanical properties, then the 
electronic transmitter is programmed with the various constant values describing those properties. 
The following photograph shows a close-up view of the nameplate on a Rosemount (Micro-Motion) 
Coriolis mass flowmeter, showing the physical constant values determined for that specific flowtube 
assembly at the time of manufacture: 

Coriolis flowmeters are equipped with RTD temperature sensors to continuously monitor the 
process fluid temperature. Fluid temperature is important to know because it affects certain 
properties of the tubes (e.g. spring constant, diameter, and length). The temperature indication is 
usually accessible as an auxiliary output, which means a Coriolis flowmeter may double as a (very 
expensive!) temperature transmitter. 

Another variable is measured and (potentially) transmitted by a Coriolis flowmeter, and this 
variable is fluid density. The tubes within a Coriolis flowmeter are shaken at their mechanical 
resonant frequency to maximize their shaking motion with the least amount of applied power to 
the force coil possible. The electronics module continuously varies the force coil's AC excitation 
frequency to maintain mechanical resonance. This resonant frequency happens to change with 
process fluid density, since the effective mass of the fluid-filled tubes changes with process fluid 
density 30 , and mass is one of the variables influencing the resonant frequency of any physical object. 

30 If you consider each tube as a container with a fixed volume capacity, a change in fluid density (e.g. pounds per 
cubic foot) must result in a change of weight for each tube. 



Note the "mass" term in the following formula, describing the resonant frequency of a tensed string: 


/ = Fundamental resonant frequency of string (Hertz) 

L = String length (meters) 

Ft = String tension (newtons) 

/i = Unit mass of string (kilograms per meter) 

This means fluid density, along with fluid temperature, is another variable measured by a Coriolis 
flowmeter. The ability to simultaneously measure these three variables (mass flow rate, temperature, 
and density) makes the Coriolis flowmeter a very versatile instrument indeed. This is especially true 
when the flowmeter in question communicates digitally using a "fieldbus" standard rather than an 
analog 4-20 mA signal. Fieldbus communication allows multiple variables to be transmitted by the 
device to the host system (and/or to other devices on the same fieldbus network), allowing the 
Coriolis flowmeter to do the job of three instruments! 

An example of a Coriolis mass flowmeter being used as a multi- variable transmitter appears in 
the following photographs. Note the instrument tag labels in the close-up photograph (FT, TT, and 
DT), documenting its use as a flow transmitter, temperature transmitter, and density transmitter, 

B ■ i ,jI 



/■'^L jM ''■ 

§m i 


r \ 

l i r 

Even though a Coriolis flowmeter inherently measures mass flow rate, the continuous 
measurement of fluid density allows the meter to calculate volumetric flow rate if this is the preferred 
means of expressing fluid flow. The relationship between mass flow (W), volumetric flow (Q), and 
mass density (p) is quite simple: 

W = pQ 



All the flowmeter's computer must do to output a volumetric flow measurement is take the 
mass flow measurement value and divide that by the fluid's measured density. A simple exercise in 



dimensional analysis (performed with metric units of measurement) validates this concept for both 
forms of the equation shown above: 






m 3 

r 3 "i 


m :! 

Coriolis mass flowmeters are very accurate and dependable. They are also completely immune 
to swirl and other fluid disturbances, which means they may be located nearly anywhere in a 
piping system with no need at all for straight-run pipe lengths upstream or downstream of the 
meter. Their natural ability to measure true mass flow, along with their characteristic linearity 
and accuracy, makes them ideally suited for custody transfer applications (where the flow of fluid 
represents product being bought and sold) . Perhaps the greatest disadvantage of Coriolis flowmeters 
is their high initial cost, especially for large pipe sizes. 


15.6 Thermal-based (mass) flowmeters 

Wind chill is a phenomenon common to nearly everyone who has ever lived in a cold environment. 
When the ambient air temperature is substantially colder than the temperature of your body, heat 
will transfer from your body to the surrounding air. If there is no breeze to move air past your body, 
the air molecules immediately surrounding your body will begin to warm up as they absorb heat 
from your body, which will then decrease the rate of heat loss. However, if there is even a slight 
breeze of air moving past your body, your body will come into contact with more cool (unheated) 
air molecules than it would otherwise, causing a greater rate of heat loss. Thus, your perception of 
the surrounding temperature will be cooler than if there were no breeze. 

We may exploit this principle to measure mass flow rate, by placing a heated object in the midst 
of a fluid flowstream, and measuring how much heat the flowing fluid convects away from the heated 
object. The "wind chill" experienced by that heated object is a function of true mass flow rate (and 
not just volumetric flow rate) because the mechanism of heat loss is the rate at which fluid molecules 
contact the heated object, with each of those molecules having a definite mass. 

The simplest form of thermal mass flowmeter is the hot-wire anemometer, used to measure air 
speed. This flowmeter consists of a metal wire through which an electric current is passed to heat 
it up. An electric circuit monitors the resistance of this wire (which is directly proportional to wire 
temperature because most metals have a definite temperature coefficient of resistance) . If air speed 
past the wire increases, more heat will be drawn away from the wire and cause its temperature to 
drop. The circuit senses this temperature change and compensates by increasing current through 
the wire to bring its temperature back up to setpoint. The amount of current sent through the wire 
becomes a representation of mass air flow rate past the wire. 

Most mass air flow sensors used in automotive engine control applications employ this principle. 
It is important for engine control computers to measure mass air flow and not just volumetric air 
flow because it is important to maintain proper air/fuel ratio even if the air density changes due 
to changes in altitude. In other words, the computer needs to know how many air molecules are 
entering the engine per second in order to properly meter the correct amount of fuel into the engine 
for complete and efficient combustion. The "hot wire" mass air flow sensor is simple and inexpensive 
to produce in quantity, which is why it finds common use in automotive applications. 

Industrial thermal mass flowmeters usually consist of a specially designed "flowtube" with two 
temperature sensors inside: one that is heated and one that is unheated. The heated sensor acts 
as the mass flow sensor (cooling down as flow rate increases) while the unheated sensor serves to 
compensate for the "ambient" temperature of the process fluid. A typical thermal mass flowtube 
appears in the following diagrams (note the swirl vanes in the close-up photograph, designed to 
introduce large-scale turbulence into the flowstream to maximize the convective cooling effect of the 
fluid against the heated sensor element): 



The simple construction of thermal mass flowmeters allows them to be manufactured in very 
small sizes. The following photograph shows a small device that is not only a mass flow meter, but 
also a mass flow controller with its own built-in throttling valve mechanism and control electronics. 
To give you a sense of scale, the tube fittings seen on the left- and right-hand sides of this device 
are 1/4 inch, making this photograph nearly full-size: 


An important factor in the calibration of a thermal mass flowmeter is the specific heat of the 
process fluid. "Specific heat" is a measure of the amount of heat energy needed to change the 
temperature of a standard quantity of substance by some specified amount 31 . Some substances have 
much greater specific heat values than others, meaning those substances have the ability to absorb 
(or release) a lot of heat energy without experiencing a great temperature change. Fluids with high 
specific heat values make good coolants, because they are able to remove much heat energy from 
hot objects without experiencing great increases in temperature themselves. Since thermal mass 
flowmeters work on the principle of convective cooling, this means a fluid having a high specific heat 
value will elicit a greater response from a thermal mass flowmeter than the exact same mass flow 
rate of a fluid having a lesser specific heat value (i.e. a fluid that is not as good of a coolant). 

This means we must know the specific heat value of whatever fluid we plan to measure with a 
thermal mass flowmeter, and we must be assured its specific heat value will remain constant. For 
this reason, thermal mass flowmeters are not suitable for measuring the flow rates of fluid streams 
whose chemical composition is likely to change over time. This limitation is analogous to that of 
a pressure sensor used to hydrostatically measure the level of liquid in a vessel: in order for this 
level-measurement technique to be accurate, we must know the density of the liquid and also be 
assured that density will be constant over time. 

31 For example, the specific heat of water is 1.00 kcal / kg • C°, meaning that the addition of 1000 calories of heat 
energy is required to raise the temperature of 1 kilogram of water by 1 degree Celsius, or that we must remove 1000 
calories of heat energy to cool that same quantity of water by 1 degree Celsius. Ethyl alcohol, by contrast, has a 
specific heat value of only 0.58 kcal / kg • C°, meaning it is almost twice as easy to warm up or cool down as water 
(little more than half the energy required to heat or cool water needs to be transferred to heat or cool the same mass 
quantity of ethyl alcohol by the same amount of temperature). 



15.7 Positive displacement flowmeters 

A positive displacement flowmeter is a cyclic mechanism built to pass a fixed volume of fluid through 
with every cycle. Many positive displacement flowmeters are rotary in nature, meaning each shaft 
revolution represents a certain volume of fluid has passed through the meter. 

Positive displacement flowmeters have been the traditional choice for residential and commercial 
natural gas flow and water flow measurement in the United States (a simple application of custody 
transfer flow measurement, where the fluid being measured is a commodity bought and sold). The 
cyclic nature of a positive displacement meter lends itself well to total gas quantity measurement 
(and not just flow rate), as the mechanism may be coupled to a mechanical counter which is read by 
utility personnel on a monthly basis. A rotary gas flowmeter is shown in the following photograph. 
Note the odometer-style numerical display on the left-hand end of the meter, totalizing gas usage 
over time: 

Positive displacement flowmeters rely on moving parts to shuttle quantities of fluid through them, 
and these moving parts must effectively seal against each other to prevent leakage past the mechanism 
(which will result in the instrument indicating less fluid passing through than there actually is) . The 
finely-machined construction of a positive displacement flowmeter will suffer damage from grit or 
other abrasive materials present in the fluid, which means these flowmeters are applicable only 
to clean fluid flowstreams. Even with clean fluid flowing through, the mechanisms are subject to 
wear and accumulating inaccuracies over time. However, there is really nothing more definitive for 
measuring volumetric flow rate than an instrument built to measure individual volumes of fluid with 
each mechanical cycle. As one might guess, these instruments are completely immune to swirl and 



other large-scale fluid turbulence, and may be installed nearly anywhere in a piping system (no need 
for long sections of straight- length pipe upstream or downstream) . Positive displacement flowmeters 
are also very linear, since mechanism cycles are directly proportional to fluid volume. 

15.8 Weighfeeders 

A completely different kind of flowmeter is the weighfeeder, used to measure the flow of solid material 
such as powders and grains. One of the most common weighfeeder designs consists of a conveyor 
belt with a section supported by rollers coupled to one or more load cells, such that a fixed length 
of the belt is continuously weighed: 

Material from 
storage bin 

Feed chute 

Solid powder or granules 

Belt motion To pr0 cess 

The load cell measures the weight of a fixed-length belt section, yielding a figure of material 
weight per linear distance on the belt. A tachometer (speed sensor) measures the speed of the belt. 
The product of these two variables is the mass flow rate of solid material "through" the weighfeeder: 




W = Mass flow rate (e.g. pounds per second) 

F = Force of gravity acting on the weighed belt section (e.g. pounds) 

S = Belt speed (e.g. feet per second) 

d = Length of weighed belt section (e.g. feet) 



15.9 Change-of-quantity flow measurement 

Flow, by definition, is the passage of material from one location to another over time. So far 
this chapter has explored technologies for measuring flow rate en route from source to destination. 
However, a completely different method exists for measuring flow rates: measuring how much 
material has either departed or arrived at the terminal locations over time. 

Mathematically, we may express flow as a ratio of quantity to time. Whether it is volumetric flow 
or mass flow we are referring to, the concept is the same: quantity of material moved per quantity 
of time. We may express average flow rates as ratios of changes: 

W = 


W = Average mass flow rate 

Q = Average volumetric flow rate 

Am = Change in mass 

AV = Change in volume 

At = Change in time 




Suppose a water storage vessel is equipped with load cells to precisely measure weight (which is 
directly proportional to mass with constant gravity). Assuming only one pipe entering or exiting 
the vessel, any flow of water through that pipe will result in the vessel's total weight changing over 

///////////// 7 7 //////////////// 7 7 ///////// / 


If the measured mass of this vessel decreased from 74,688 kilograms to 70,100 kilograms between 
4:05 AM and 4:07 AM, we could say that the average mass flow rate of water leaving the vessel is 
2,294 kilograms per minute over that time span. 

— _ Am _ 70100 kg - 74688 kg _ -4588 kg _ kg 

At ~~ 4:07 - 4:05 2 min ~ min 

Note that this average flow measurement may be determined without any flowmeter of any kind 
installed in the pipe to intercept the water flow. All the concerns of flowmeters studied thus far 
(turbulence, Reynolds number, fluid properties, etc.) are completely irrelevant. We may measure 
practically any flow rate we desire simply by measuring stored weight (or volume) over time. A 
computer may do this calculation automatically for us if we wish, on practically any time scale 

Now suppose the practice of determining average flow rates every two minutes was considered 
too infrequent. Imagine that operations personnel require flow data calculated and displayed more 
often than just 30 times an hour. All we must do to achieve better time resolution is take weight 
(mass) measurements more often. Of course, each mass-change interval will be expected to be less 
with more frequent measurements, but the amount of time we divide by in each calculation will be 
proportionally smaller as well. If the flow rate happens to be absolutely steady, we may sample mass 
as frequently as we might like and we will still arrive at the same flow rate value as before (sampling 
mass just once every two minutes). If, however, the flow rate is not steady, sampling more often will 
allow us to better see the immediate "ups" and "downs" of flow behavior. 

Imagine now that we had our hypothetical "flow computer" take weight (mass) measurements at 
an infinitely fast pace: an infinite number of samples per second. Now, we are no longer averaging 
flow rates over finite periods of time; instead we would be calculating instantaneous flow rate at any 
given point in time. 

Calculus has a special form of symbology to represent such hypothetical scenarios: we replace 
the Greek letter "delta" (A, meaning "change") with the roman letter "d" (meaning differential). 
A simple way of picturing the meaning of "d" is to think of it as meaning an infinitesimal interval of 
whatever variable follows the "d" in the equation 32 . When we set up two differentials in a quotient, 
we call the 4 fraction a derivative. Re-writing our average flow rate equations in derivative (calculus) 


W = Instantaneous mass flow rate 

Q = Instantaneous volumetric flow rate 

dm = Infinitesimal (infinitely small) change in mass 

dV = Infinitesimal (infinitely small) change in volume 

dt = Infinitesimal (infinitely small) change in time 

We need not dream of hypothetical computers capable of infinite calculations per second in order 
to derive a flow measurement from a mass (or volume) measurement. Analog electronic circuitry 

32 While this may seem like a very informal definition of differential, it is actually rooted in a field of mathematics 
called nonstandard analysis, and closely compares with the conceptual notions envisioned by calculus' founders. 



exploits the natural properties of resistors and capacitors to essentially do this very thing in real 


33 . 

Differentiator circuit 

Voltage signal 

in from mass 


r O 


O -i 

Voltage signal 
out representing 
mass flow rate 



In the vast majority of applications you will see digital computers used to calculate average 
flow rates rather than analog electronic circuits calculating instantaneous flow rates. The 
broad capabilities of digital computers virtually ensures they will be used somewhere in the 
measurement/control system, so the rationale is to use the existing digital computer to calculate 
flow rates (albeit imperfectly) rather than complicate the system design with additional (analog) 
circuitry. As fast as modern digital computers are able to process simple calculations such as these 
anyway, there is little practical reason to prefer analog signal differentiation except in specialized 
applications where high speed performance is paramount. 

Perhaps the single greatest disadvantage to inferring flow rate by differentiating mass or volume 
measurements over time is the requirement that the storage vessel have but one flow path in and 
out. If the vessel has multiple paths for liquid to move in and out (simultaneously), any flow rate 
calculated on change-in-quantity will be a net flow rate only. It is impossible to use this flow 
measurement technique to measure one flow out of multiple flows common to one liquid storage 

A simple "thought experiment" confirms this fact. Imagine a water storage vessel receiving a flow 
rate in at 200 gallons per minute. Next, imagine that same vessel emptying water out of a second 
pipe at the exact same flow rate: 200 gallons per minute. With the exact same flow rate both entering 
and exiting the vessel, the water level in the vessel will remain constant. Any change-of-quantity 
flow measurement system would register zero change in mass or volume over time, consequently 
calculating a flow rate of absolutely zero. Truly, the net flow rate for this vessel is zero, but this 
tells us nothing about the flow in each pipe, except that those flow rates are equal in magnitude and 
opposite in direction. 

33 To be precise, the equation describing the function of this analog differentiator circuit is: V ou t = —RC ,'" . The 
negative sign is an artifact of the circuit design - being essentially an inverting amplifier with negative gain - and not 
an essential element of the math. 


15.10 Insertion flowmeters 

This section does not describe a particular type of flowmeter, but rather a design that may be 
implemented for several different kinds of flow measurement technologies. When the pipe carrying 
process fluid is large in size, it may be impractical or cost-prohibitive to install a full-diameter 
flowmeter to measure fluid flow rate. A practical alternative for many applications is the installation 
of an insertion flowmeter: a probe that may be inserted into or extracted from a pipe, to measure 
fluid velocity in one region of the pipe's cross-sectional area (usually the center). 

A classic example of an insertion flowmeter element is the Annubar, a form of averaging pitot 
tube pioneered by the Dieterich Standard corporation. The Annubar flow element is inserted into a 
pipe carrying fluid where it generates a differential pressure for a pressure sensor to measure: 


Ball valve 

Pipe wall 


Compression nut 
("Gland" nut) 


Pipe wall 

Pipe wall 

The Annubar element may be extracted from the pipe by loosening a "gland nut" and pulling the 
assembly out until the end passes through a hand ball valve. Once the element has been extracted 
this far, the ball valve may be shut and the Annubar completely removed from the pipe: 



Loosen this nut to 
\ /extract the Annubar 

Ball valve 

Pipe wall 

Close this ball valve 
when the Annubar is clear 

Pipe wall 


Pipe wall 

Other flowmeter technologies manufactured in insertion form include vortex, turbine, and thermal 
mass. If the flow-detection element is compact rather than distributed, care must be taken to ensure 
correct positioning within the pipe. Since flow profiles are never completely flat, any insertion meter 
element will register a greater flow rate at the center of the pipe than near the walls. Wherever the 
insertion element is placed in the pipe diameter, that placement must remain consistent through 
repeated extractions and re-insertions or else the effective calibration of the insertion flowmeter will 
change every time it is removed and re-inserted into the pipe. Care must also be taken to insert the 
flowmeter so that the flow element points directly upstream, and not at an angle. 

A unique advantage of insertion instruments is that they may be installed in an operating pipe 
by using specialized hot-tapping equipment. A "hot tap" is a procedure whereby a safe penetration 
is made into a pipe while the pipe is carrying fluid under pressure. The first step in a hot-tapping 
operation is to weld a "saddle tee" fitting on the side of the pipe: 







Next, a ball valve is bolted onto the saddle tee flange. This ball valve will be used to isolate the 
insertion instrument from the fluid pressure inside the pipe: 

Ball valve 



A special hot-tapping drill is then bolted to the open end of the ball valve. This drill uses a 
high-pressure seal to contain fluid pressure inside the drill chamber as a motor spins the drill bit. 
The ball valve is opened, then the drill bit is advanced toward the pipe wall where it cuts a hole 
into the pipe. Fluid pressure rushes into the empty chamber of the ball valve and hot-tapping drill 
as soon as the pipe wall is breached: 



drill motor 


High-pressure seals 

Ball valve 
(open) . 

Once the hole has been completely drilled, the bit is extracted and the ball valve shut to allow 
removal of the hot-tapping drill: 



drill motor 

Ball valve 



Now there is a flanged and isolated connection into the "hot" pipe, through which an insertion 
flowmeter (or other instrument/device) may be installed. 

Hot-tapping is a technical skill, with many safety concerns specific to different process fluids, 
pipe types, and process applications. This brief introduction to the technique is not intended to be 
instructional, but merely informational. 


15.11 Process/instrument suitability 

Every flow-measuring instrument exploits a physical principle to measure the flow rate of fluid 
stream. Understanding each of these principles as they apply to different flow-measurement 
technologies is the first and most important step in properly applying a suitable technology to 
the measurement of a particular process stream flow rate. The following table lists the specific 
operating principles exploited by different flow measurement technologies: 

Flow measurement 




Differential pressure 

Fluid mass self-acceleration, 
potential-kinetic energy exchange 




Viscous fluid friction 



Weirs & flumes 

Fluid mass self-acceleration, 
potential-kinetic energy exchange 

H n 


Turbine (velocity) 

Fluid velocity spinning 
a vaned wheel 




von Karman effect 




Electromagnetic induction 




Sound wave time-of-flight 




Fluid inertia, 
Coriolis effect 



Turbine (mass) 

Fluid inertia 




Convective cooling, 
specific heat of fluid 



Positive displacement 

Movement of fixed volumes 



A potentially important factor in choosing an appropriate flowmeter technology is energy loss 
caused by pressure drop. Some flowmeter designs, such as the common orifice plate, are inexpensive 
to install but carry a high price in terms of the energy lost in permanent pressure drop. Energy 
costs money, and so industrial facilities would be wise to consider the long-term cost of a flowmeter 
before settling on the one that is cheapest to install. It could very well be, for example, that an 
expensive venturi tube will cost less after years of operation than a cheap orifice plate. 

In this regard, certain flowmeters stand above the rest: those with obstructionless flowtubes. 
Magnetic and ultrasonic flowmeters have no obstructions whatsoever in the path of the flow. This 
translates to (nearly) zero permanent pressure loss along the length of the tube, and therefore. 
Thermal mass and straight-tube Coriolis flowmeters are nearly obstructionless, while vortex and 
turbine meters are only slightly worse. 



AG A Report No. 3 - Orifice metering of natural gas and other related hydrocarbon fluids, 
Part 1 (General Equations and Uncertainty Guidelines), Catalog number XQ9017, American Gas 
Association and American Petroleum Institute, Washington D.C., Third Edition October 1990, 
Second Printing June 2003. 

AG A Report No. 3 - Orifice metering of natural gas and other related hydrocarbon fluids, Part 2 
(Specification and Installation Requirements), Catalog number XQ0002, American Gas Association 
and American Petroleum Institute, Washington D.C., Fourth Edition April 2000, Second Printing 
June 2003. 

AG A Report No. 3 - Orifice metering of natural gas and other related hydrocarbon fluids, Part 
3 (Natural Gas Applications), Catalog number XQ9210, American Gas Association and American 
Petroleum Institute, Washington D.C., Third Edition August 1992, Second Printing June 2003. 

AG A Report No. 3 - Orifice metering of natural gas and other related hydrocarbon fluids, 
Part 4 (Background, Development, Implementation Procedure, and Subroutine Documentation for 
Empirical Flange-Tapped Discharge Coefficient Equation), Catalog number XQ9211, American Gas 
Association and American Petroleum Institute, Washington D.C., Third Edition October 1992, 
Second Printing August 1995, Third Printing June 2003. 

Chow, Ven Te., Open-Channel Hydraulics, McGraw-Hill Book Company, Inc., New York, NY, 1959. 

"Flow Measurement User Manual", Form Number A6043, Part Number D301224X012, Emerson 
Process Management, 2005. 

Fribance, Austin E., Industrial Instrumentation Fundamentals, McGraw-Hill Book Company, New 
York, NY, 1962. 

General Specifications: "EJX910A Multivariable Transmitter", Document GS 01C25R01-01E, 5th 
edition, Yokogawa Electric Corporation, Tokyo, Japan, 2005. 

Giancoli, Douglas C, Physics for Scientists & Engineers, Third Edition, Prentice Hall, Upper Saddle 
River, New Jersey, 2000. 

Hofmann, Friedrich, Fundamentals of Ultrasonic Flow Measurement for industrial applications, 
Krohne Messtechnik GmbH & Co. KG, Duisburg, Germany, 2000. 

Hofmann, Friedrich, Fundamental Principles of Electromagnetic Flow Measurement, 3rd Edition, 
Krohne Messtechnik GmbH & Co. KG, Duisburg, Germany, 2003. 

Kallen, Howard P., Handbook of Instrumentation and Controls, McGraw-Hill Book Company, Inc., 
New York, NY, 1961. 

Keisler, H. Jerome, Elementary Calculus - An Infinitesimal Approach, Second Edition, University 
of Wisconsin, 2000. 


Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Miller, Richard W., Flow Measurement Engineering Handbook, Second Edition, McGraw-Hill 
Publishing Company, New York, NY, 1989. 

Price, James F., A Coriolis Tutorial, version 3.3, Woods Hole Oceanographic Institution, Woods 
Hole, MA, 2006. 

Spink, L. K., Principles and Practice of Flow Meter Engineering, Ninth Edition, The Foxboro 
Company, Foxboro, MA, 1967. 

Vennard, John K., Elementary Fluid Mechanics, 3rd Edition, John Wiley & Sons, Inc., New York, 
NY, 1954. 


Chapter 16 

Continuous analytical 


16.1 Density measurement 

16.2 Turbidity measurement 



16.3 Conductivity measurement 

Electrical conductivity in metals is the result of free electrons drifting within a "lattice" of atomic 
nuclei comprising the metal object. When a voltage is applied across two points of a metal object, 
these free electrons immediately drift toward the positive pole (anode) and away from the negative 
pole (cathode). 

Electrical conductivity in liquids is another matter entirely. Here, the charge carriers are ions: 
electrically imbalanced atoms or molecules that are free to drift because they are not "locked" into 
a lattice structure as is the case with solid substances. The degree of electrical conductivity of any 
liquid is therefore dependent on the ion density of the solution (how many ions freely exist per 
unit volume of liquid). When a voltage is applied across two points of a liquid solution, negative 
ions will drift toward the positive pole (anode) and positive ions will drift toward the negative pole 
(cathode). In honor of this directional drifting, negative ions are sometimes called anions (attracted 
to the anode), while positive ions are sometimes called cations (attracted to the cathode). 

Electrical conductivity in gases is much the same: ions are the charge carriers. However, with 
gases at room temperature, ionic activity is virtually nonexistent. A gas must be superheated into 
a plasma state before substantial ions exist which can support an electric current. 

16.3.1 Dissociation and ionization in aqueous solutions 

Pure water is a very poor conductor of electricity. Some water molecules will "ionize" into unbalanced 
halves (instead of H2O, you will find some negatively charged hydroxyl ions (OH - ) and some 
positively charged hydrogen ions 1 (H + ), but the percentage is extremely small at room temperature. 

Any substance that enhances electrical conductivity when dissolved in water is called an 
electrolyte. This enhancement of conductivity occurs due to the molecules of the electrolyte 
separating into positive and negative ions, which are then free to serve as electrical charge carriers. 
If the electrolyte in question is an ionically-bonded compound 2 (table salt is a common example), the 
ions forming that compound naturally separate in solution, and this separation is called dissociation. 
If the electrolyte in question is a covalently-bonded compound 3 (hydrogen chloride is an example), 
the separation of those molecules into positive and negative ions is called ionization. 

Both dissociation and ionization refer to the separation of formerly joined atoms upon entering a 
solution. The difference between these terms is the type of substance that splits: "dissociation" refers 
to the division of ionic compounds (such as table salt) , while "ionization" refers to covalent-bonded 
(molecular) compounds such as HC1 which are not ionic in their pure state. 

Ionic impurities added to water (such as salts and metals) immediately dissociate and become 
available to act as charge carriers. Thus, the measure of a water sample's electrical conductivity is 

1 Truth be told, free hydrogen ions are extremely rare in an aqueous solution. You are far more likely to find them 
bound to normal water molecules to form positive hydronium ions (Ji3 + ). For simplicity's sake, though, professional 
literature often refers to these positive ions as "hydrogen" ions and even represent them symbolically as H+. 

2 Ionic compounds are formed when oppositely charged atomic ions bind together by mutual attraction. The 
distinguishing characteristic of an ionic compound is that it is a conductor of electricity in its pure, liquid state. That 
is, it readily separates into anions and cations all by itself. Even in its solid form, an ionic compound is already 
ionized, with its constituent atoms held together by an imbalance of electric charge. Being in a liquid state simply 
gives those atoms the physical mobility needed to dissociate. 

3 Covalent compounds are formed when neutral atoms bind together by the sharing of valence electrons. Such 
compounds are not good conductors of electricity in their pure, liquid states. 



a fair estimate of ionic impurity concentration. Conductivity is therefore an important analytical 
measurement for certain water purity applications, such as the treatment of boiler feedwater, and 
the preparation of high-purity water used for semiconductor manufacturing. 

It should be noted that conductivity measurement is a very non-specific form of analytical 
measurement. The conductivity of a liquid solution is a gross indication of its ionic content, but it 
tells us nothing specific about the type or types of ions present in the solution. Therefore, conductivity 
measurement is meaningful only when we have prior knowledge of the particular ionic species present 
in the solution (or when the purpose is to eliminate all ions in the solution such as in the case of 
ultra-pure water treatment, in which case we do not care about types of ions because our ideal goal 
is zero conductivity). 

16.3.2 Two-electrode conductivity probes 

Conductivity is measured by an electric current passed through the solution. The most primitive 
form of conductivity sensor (sometimes referred to as a conductivity cell) consists of two metal 
electrodes inserted in the solution, connected to a circuit designed to measure conductance (G), the 
reciprocal of resistance ( -p): 


source Ammeter 

V I 

Distance = d 

h H 

G = 







Area = A 

The conductance measured by this instrument is a function of plate geometry (surface area and 
distance of separation) as well as the ionic activity of the solution. A simple increase in separation 
distance between the probe electrodes will result in a decreased conductance measurement (increased 
resistance R) even if the liquid solution's ionic properties do not change. Therefore, conductance 
(G) is not particularly useful as an expression of liquid conductivity. 

The mathematical relationship between conductance (G), plate area (A), plate distance (ci), and 
the actual conductivity of the liquid (k) is expressed in the following equation 4 : 

G = k- 


4 This equation bears a striking similarity to the equation for resistance of metal wire: R = p^-, where / is the 
length of a wire sample, A is the cross-sectional area of the wire, and p is the specific resistance of the wire metal. 



G = Conductance, in Siemens (S) 

k = Specific conductivity of liquid, in Siemens per centimeter (S/cm) 

A = Electrode area (each), in square centimeters (cm 2 ) 

d = Electrode separation distance, in centimeters (cm) 

The unit of Siemens per centimeter may seem odd at first, but it is necessary to account for all 
the units present in the variables of the equation. A simple dimensional analysis proves this: 




For any particular conductivity cell, the geometry may be expressed as a ratio of separation 
distance to plate area, usually symbolized by the lower-case Greek letter Theta (9), and always 
expressed in the unit of inverse centimeters (cm -1 ): 

Re-writing the conductance equation using 9 instead of A and d, we see that conductance is the 
quotient of conductivity k and the cell constant 9: 

G- k 


G = Conductance, in Siemens (S) 

k = Specific conductivity of liquid, in Siemens per centimeter (S/cm) 

9 = Cell constant, in inverse centimeters (cm -1 ) 

Manipulating this equation to solve for conductivity (fc) given electrical conductance (G) and 
cell constant (0), we have the following result: 

k = G9 

Two-electrode conductivity cells are not very practical in real applications, because mineral 
and metal ions attracted to the electrodes tend to "plate" the electrodes over time forming solid, 
insulating barriers on the electrodes. While this "electroplating" action may be substantially reduced 
by using AC instead of DC 5 to excite the sensing circuit, it is usually not enough. Over time, the 
conductive barriers formed by ions bonded to the electrode surfaces will create calibration errors by 
making the instrument "think" the liquid is less conductive than it actually is. 

16.3.3 Four-electrode conductivity probes 

A very old electrical technique known as the Kelvin or four-wire resistance-measuring method is a 
practical solution for this problem. Commonly employed to make precise resistance measurements 

B The use of alternating current forces the ions to switch directions of travel many times per second, thus reducing 
the chance they have of bonding to the metal electrodes. 



for scientific experiments in laboratory conditions, as well as measuring the electrical resistance of 
strain gauges and other resistive sensors, the four-wire technique uses four conductors to connect 
the resistance under test to the measuring instrument: 

4-wire ohmmeter 


4-wire cable 

, vW 


(wire resistance) 




Voltmeter indication 
Current source 

Only the outer two conductors carry substantial current. The inner two conductors connecting 
the voltmeter to the test specimen carry negligible current (due to the voltmeter's extremely high 
input impedance) and therefore drop negligible voltage along their lengths. Voltage dropped across 
the current-carrying (outer) wires is irrelevant, since that voltage drop is never detected by the 

Since the voltmeter only measures voltage dropped across the specimen (the resistor under test), 
and not the test resistance plus wiring resistance, the resulting resistance measurement is much 
more accurate. In the case of conductivity measurement, it is not wire resistance that we care to 
ignore, but rather the added resistance caused by plating of the electrodes. By using four electrodes 
instead of two, we are able to measure voltage dropped across a length of liquid solution only, and 
completely ignore the resistive effects of electrode plating: 


/ Sample / 
v liquid ' 



In the 4-wire conductivity cell, any electrode plating will merely burden the current source by 
causing it to output a greater voltage, but it will not affect the amount of voltage detected by the two 
inner electrodes as that electric current passes through the liquid. Some conductivity instruments 
employ a second voltmeter to measure the voltage dropped between the "excitation" electrodes, to 
indicate electrode fouling: 



(Second) voltmeter 












Any form of electrode fouling will cause this secondary voltage measurement to rise, thus 
providing an indicator that instrument technicians may use for predictive maintenance (telling them 
when the probes need cleaning or replacement). Meanwhile, the primary voltmeter will do its job 
of accurately measuring liquid conductivity so long as the current source is still able to output its 
normal amount of current. 

16.3.4 Electrodeless conductivity probes 

An entirely different design of conductivity cell called electrodeless uses electromagnetic induction 
rather than direct electrical contact to detect the conductivity of the liquid solution. This cell design 
enjoys the distinct advantage of virtual immunity to fouling 6 , since there is no direct electrical contact 
between the measurement circuit and the liquid solution. Instead of using two or four electrodes 
inserted into the solution for conductivity measurement, this cell uses two toroidal inductors (one to 
induce an AC voltage in the liquid solution, and the other to measure the strength of the resulting 
current through the solution): 

Toroidal conductivity sensors may suffer calibration errors if the fouling is so bad that the hole becomes choked 
off with sludge, but this is an extreme condition. These sensors are far more tolerant to fouling than any form of 
contact-type (electrode) conductivity cell. 



AC voltage AC 

source voltmeter 


Since toroidal magnetic cores do an excellent job of containing their own magnetic fields, there 
will be negligible mutual inductance between the two wire coils. The only way a voltage will be 
induced in the secondary coil is if there is an AC current passing through the center of that coil, 
through the liquid itself. The primary coil is ideally situated to induce such a current in the solution. 
The more conductive the liquid solution, the more current will pass through the center of both coils 
(through the liquid), thus producing a greater induced voltage at the secondary coil. Secondary coil 
voltage therefore is directly proportional to liquid conductivity 7 . 

The equivalent electrical circuit for a toroidal conductivity probe looks like a pair of transformers, 
with the liquid acting as a resistive path for current to connect the two transformers together: 




AC voltage (W\ 
source \jy 

V ) AC 
ZJ voltmeter 




Toroidal conductivity cells are whenever possible, due to their ruggedness and virtual immunity 
to fouling. However, they are not sensitive enough for conductivity measurement in high-purity 
applications such as boiler feedwater treatment and ultra-pure water treatment necessary for 
pharmaceutical and semiconductor manufacturing. As always, the manufacturer's specifications 
are the best source of information for conductivity cell applicability in any particular process. 

The following photograph shows a toroidal conductivity probe along with a conductivity 
transmitter (to both display the conductivity measurement in millisiemens per centimeter and also 
transmit the measurement as a 4-20 mA analog signal): 

7 Note that this is opposite the behavior of a direct-contact conductivity cell, which produces less voltage as the 
liquid becomes more conductive. 





16.4 pH measurement 

pH is the measurement of the hydrogen ion activity in a liquid solution. It is one of the most common 
forms of analytical measurement in industry, because pH has a great effect on the outcome of many 
chemical processes. Food processing, water treatment, pharmaceutical production, steam generation 
(thermal power plants), and alcohol manufacturing are just some of the industries making extensive 
use of pH measurement (and control). pH is also a significant factor in the corrosion of metal pipes 
and vessels carrying aqueous (water-based) solutions, so pH measurement and control is important 
in the life-extension of these capital investments. 

In order to understand pH measurement, you must first understand the chemistry of pH. Please 
refer to section 2.7 beginning on page 69 for a theoretical introduction to pH. 

16.4.1 Colorimetric pH measurement 

One of the simplest ways to measure the pH of a solution is by color. Certain specific chemicals 
dissolved in an aqueous solution will change color if the pH value of that solution falls within a 
certain range. Litmus paper is a common laboratory application of this principle, where a color- 
changing chemical substance infused on a paper strip changes color when dipped in the solution. 
Comparing the final color of the litmus paper to a reference chart yields an approximate pH value 
for the solution. 

A natural example of this phenomenon is well- know to flower gardeners, who recognize that 
hydrangea blossoms change color with the pH value of the soil. In essence, these plants act as 
organic litmus indicators 8 . This hydrangea plant indicates acidic soil by the violet color of its 

8 Truth be told, the color of a hydrangea blossom is only indirectly determined by soil pH. Soil pH affects the plant's 
uptake of aluminum, which is the direct cause of color change. Interesting, the pH-color relationship of a hydrangea 
plant is exactly opposite that of common laboratory litmus paper: red litmus paper indicates an acidic solution while 
blue litmus paper indicates an alkaline solution; whereas red hydrangea blossoms indicate alkaline soil while blue (or 
violet) hydrangea blossoms indicate acidic soil. 


16.4.2 Potentiometric pH measurement 

Color-change is a common pH test method used for manual laboratory analyses, but it is not well- 
suited to continuous process measurement. By far the most common pH measurement method in 
use is electrochemical: special pH-sensitive electrodes inserted into an aqueous solution will generate 
a voltage dependent upon the pH value of that solution. 

Like all other potentiometric (voltage-based) analytical measurements, electrochemical pH 
measurement is based on the Nernst equation, which describes the electrical potential generated 
by a difference in ionic concentration between two different solutions separated by an ion-permeable 


V = Voltage produced across membrane due to ion exchange, in volts (V) 

R = Universal gas constant (8.315 J/mol-K) 

T = Absolute temperature, in Kelvin (K) 

n = Number of electrons transferred per ion exchanged (unitless) 

F = Faraday constant, in coulombs per mole (96,485 C/mol e _ ) 

C\ = Concentration of ion in measured solution, in moles per liter of solution (M) 

Ci = Concentration of ion in reference solution (on other side of membrane) , in moles per liter 
of solution (M) 

We may also write the Nernst equation using of common logarithms instead of natural logarithms, 
which is usually how we see it written in the context of pH measurement: 

T , 2.303RT , fC x 

In the case of pH measurement, the Nernst equation describes the amount of electrical voltage 
developed across a special glass membrane due to hydrogen ion exchange between the process liquid 
solution and a buffer solution inside the bulb formulated to maintain a constant pH value of 7.0 pH. 
Special pH-measurement electrodes are manufactured with a closed end made of this glass, with the 
buffer solution contained within the glass bulb: 



wire connection point 



Bulb filled with 
potassium chloride 
"buffer" solution 
(7.0 pH) 

.glass body 


silver chloride 

+ + + 

Very thin glass bulb, 

Voltage produced 

across thickness of 

glass membrane 

Any concentration of hydrogen ions in the process solution differing from the hydrogen ion 
concentration in the buffer solution ([H + ] = 1 x 10~ 7 M) will cause a voltage to develop across the 
thickness of the glass. Thus, a standard pH measurement electrode produces no potential when the 
process solution's pH value is exactly 7.0 pH (equal in hydrogen ion activity to the buffer solution 
trapped within the bulb). 

Actually measuring this voltage, however, presents a bit of a problem: while we have a convenient 
electrical connection to the solution inside the glass bulb, we do not have any place to connect the 



other terminal of a sensitive voltmeter to the solution outside the bulb 9 . In order to establish a 
complete circuit from the glass membrane to the voltmeter, we must create a zero-potential electrical 
junction with the process solution. To do this, we use another special electrode called a reference 

wire connection point 


Filled with 
potassium chloride 
"buffer" solution 


silver chloride 

Glass or plastic body 

B V ■ 

Porous junction 

Together, the measurement and reference electrodes provide a voltage-generating element 
sensitive to the pH value of whatever solution they are submerged in: 

3 Remember that voltage is always measured between two points! 






The most common configuration for modern pH probe sets is what is called a combination 
electrode, which combines both the glass measurement electrode and the porous reference electrode 
in a single unit. This photograph shows a typical industrial combination pH electrode: 


The red-colored plastic cap on the right-hand end of this combination electrode covers and 
protects a gold-plated coaxial electrical connector, to which the voltage-sensitive pH indicator (or 
transmitter) attaches. 



Another model of pH probe appears in the next photograph. Here, there is no protective plastic 
cap covering the probe connector, allowing a view of the gold-plated connector bars: 

A close-up photograph of the probe tip reveals the glass measurement bulb, a weep hole for 
process liquid to enter the reference electrode assembly (internal to the white plastic probe body), 
and a metal solution ground electrode: 



It is extremely important to always keep the glass electrode wet. Its proper operation depends 
on complete hydration of the glass, which allows hydrogen ions to penetrate the glass and develop 
the Nernst potential. The probes shown in these photographs are shown in a dry state only because 
they have already exhausted their useful lives and cannot be damaged any further by dehydration. 

The process of hydration - so essential to the working of the glass electrode - is also a mechanism 
of wear. Layers of glass "slough" off over time if continuously hydrated, which means that glass pH 
electrodes have a limited life whether they are being used to measure the pH of a process solution 
(continuously wet) or if they are being stored on a shelf (maintained in a wet state by a small 
quantity of potassium hydroxide held close to the glass probe by a liquid-tight cap) . It is therefore 
impossible to extend the shelf life of a glass pH electrode indefinitely. 

The voltage produced by the measurement electrode (glass membrane) is quite modest. A 
calculation for voltage produced by a measurement electrode immersed in a 6.0 pH solution shows 
this. First, we must calculate hydrogen ion concentration (activity) for a 6.0 pH solution, based on 
the definition of pH being the negative logarithm of hydrogen ion molarity: 

pH = - log[H+] 

6.0 = - log[H+] 
-6.0 = log[H+] 





iog[H 4 



H H 

H H 

1 x 10" 6 M 

This tells us the concentration of hydrogen ions in the 6.0 pH solution (which is practically the 
same as hydrogen ion activity for dilute solutions). We know that the buffer solution inside the 
glass measurement bulb has a stable value of 7.0 pH (hydrogen ion concentration of 1 x 10~ 7 M, or 
0.0000001 moles per liter), so all we need to do now is plug these values in to the Nernst equation to 
see how much voltage the glass electrode should generate. Assuming a solution temperature of 25° 
C (298.15 K), and knowing that n in the Nernst equation will be equal to 1 (since each hydrogen 
ion has a single- value electrical charge): 






(1) (96485) ° g 

1 x 10" 6 M 
1 x 10" 7 M 

V = (59.17 mV) log 10 = 59.17 mV 

If the measured solution had a value of 7.0 pH instead of 6.0 pH, there would be no voltage 
generated across the glass membrane since the two solutions' hydrogen ion activities would be equal. 
Having a solution with one decade (ten times more: exactly one "order of magnitude") greater 
hydrogen ions activity than the internal buffer solution produces 59.17 millivolts at 25 degrees 
Celsius. If the pH were to drop to 5.0 (two units away from 7.0 instead of one unit), the output 
voltage would be double: 118.3 millivolts. If the solution's pH value were more alkaline than the 
internal buffer (for example, 8.0 pH), the voltage generated at the glass bulb would be the opposite 
polarity (e.g. 8.0 pH = -59.17 mV ; 9.0 pH = -118.3 mV, etc.). The following table shows the 
relationship between hydrogen ion activity, pH value, and probe voltage 10 : 

Hydrogen ion activity 

pH value 

Probe voltage (at 25° C) 

1 x 10~ 3 M = 0.001 M 

3.0 pH 

236.7 mV 

1 x 10~ 4 M = 0.0001 M 

4.0 pH 

177.5 mV 

1 x 10~ 5 M = 0.00001 M 

5.0 pH 

118.3 mV 

1 x 10~ 6 M = 0.000001 M 

6.0 pH 

59.17 mV 

1 x 10~ 7 M = 0.0000001 M 

7.0 pH 


1 x 10" 8 M = 0.00000001 M 

8.0 pH 

-59.17 mV 

1 x 10~ 9 M = 0.000000001 M 

9.0 pH 

-118.3 mV 

1 x 10~ 10 M = 0.0000000001 M 

10.0 pH 

-177.5 mV 

1 x 10" 11 M = 0.00000000001 M 

11.0 pH 

-236.7 mV 

10 The mathematical sign of probe voltage is arbitrary. It depends entirely on whether we consider the reference 
(buffer) solution's hydrogen ion activity to be C\ or C2 in the equation. Which ever way we choose to calculate this 
voltage, though, the polarity will be opposite for acidic pH values as compared to alkaline pH values 



This numerical progression is reminiscent of the Richter scale used to measure earthquake 
magnitudes, where each ten-fold (decade) multiplication of power is represented by one more 
increment on the scale (e.g. a 6.0 Richter earthquake is ten times more powerful than a 5.0 Richter 
earthquake). The logarithmic nature of the Nernst equation means that pH probes - and in fact 
all potentiometric sensors based on the same dynamic of voltage produced by ion exchange across a 
membrane - have astounding rangeability : they are capable of representing a wide range of conditions 
with a modest signal voltage span. 

Of course, the disadvantage of high rangeability is the potential for large pH measurement errors 
if the voltage detection within the pH instrument is even just a little bit inaccurate. The problem is 
made even worse by the fact that the voltage measurement circuit has an extremely high impedance 
due to the presence of the glass membrane 11 . The pH instrument measuring the voltage produced 
by a pH probe assembly must have an input impedance that is orders of magnitude greater yet, 
or else the probe's voltage signal will become "loaded down" by the voltmeter and not register 
accurately. Fortunately, modern operational amplifier circuits with field-effect transistor input stages 
are sufficient for this task 12 : 

Equivalent electrical circuit of a pH probe and instrument 

pH probe assembly 



250 Mfi 







pH instrument 


io 12 n 


The voltage sensed by the pH instrument very nearly 
equals V pH because (R glass + R ref ) « R input 

Even if we use a high-input-impedance pH instrument to sense the voltage output by the pH 
probe assembly, we may still encounter a problem created by the impedance of the glass electrode: an 
RC time constant created by the parasitic capacitance of the probe cable connecting the electrodes 
to the sensing instrument. The longer this cable is, the worse the problem becomes due to increased 

11 Glass is a very good insulator of electricity. With a thin layer of glass being an essential part of the sensor circuit, 
the typical impedance of that circuit will lie in the range of hundreds of mega-ohms! 

12 Operational amplifier circuits with field-effect transistor inputs may easily achieve input impedances in the tera- 
ohm range (1 X 10 12 Q). 



pH probe assembly 


250 MQ. 






pai asitic 

pH instrument 



io 12 n 


This time constant value may be significant if the cable is long and/or the probe resistance is 
abnormally large. Assuming a combined (measurement and reference) electrode resistance of 700 
Mil and a 30 foot length of RG-58U coaxial cable (at 28.5 pF capacitance per foot), the time constant 
will be: 


(700 x 10 6 0) ((28.5 x 10~ 12 F/ft)(30 ft)) 

(700 x 10 6 fi)(8.55 x lO" 10 F) 

r = 0.599 seconds 

Considering the simple approximation of 5 time constants being the time necessary for a first- 
order system such as this to achieve within 1% of its final value after a step-change, this means 
a sudden change in voltage at the pH probe caused by a sudden change in pH will not be fully 
registered by the pH instrument until almost 3 seconds after the event has passed! 

It may seem impossible for a cable with capacitance measured in picofarads to generate a time 
constant easily within the range of human perception, but it is indeed reasonable when you consider 
the exceptionally large resistance value of a glass pH measurement electrode. For this reason, and 
also for the purpose of limiting the reception of external electrical "noise," it is best to keep the 
cable length between pH probe and instrument as short as possible. 

When short cable lengths are simply not practical, a preamplifier module may be connected 
between the pH probe assembly and the pH instrument. Such a device is essentially a unity-gain 
(gain = 1) amplifier designed to "repeat" the weak voltage output of the pH probe assembly in 
a much stronger (i.e. lower-impedance) form so that the effects of cable capacitance will not be 
as severe. A unity-gain operational amplifier "voltage buffer" circuit illustrates the concept of a 



pH probe assembly 


pH instrument 



rv input 
10 [2 Q. 


The preamplifier does not boost the probes' voltage output at all. Rather, it serves to decrease 
the impedance (the Thevenin equivalent resistance) of the probes by providing a low-resistance 
(relatively high-current capacity) voltage output to drive the cable and pH instrument. By providing 
a voltage gain of 1, and a very large current gain, the preamplifier practically eliminates RC time 
constant problems caused by cable capacitance, and also helps reduce the effect of induced electrical 
noise. As a consequence, the practical cable length limit is extended by orders of magnitude. 

Referring back to the Nernst equation, we see that temperature plays a role in determining 
the amount of voltage generated by the glass electrode membrane. The calculations we performed 
earlier predicting the amount of voltage produced by different solution pH values all assumed the 
same temperature: 25 degrees Celsius (298.15 Kelvin). If the solution is not at room temperature, 
however, the voltage output by the pH probe will not be 59.17 millivolts per pH unit. For example, 
if a glass measurement electrode is immersed in a solution having a pH value of 6.0 pH at 70 degrees 
Celsius (343.15 Kelvin), the voltage generated by that glass membrane will be 68.11 mV rather 
than 59.17 mV as it would be at 25 degrees Celsius. That is to say, the slope of the pH-to- voltage 
function will be 68.11 millivolts per pH unit rather than 59.17 millivolts per pH unit as it was at 
room temperature. 

In order for a pH instrument to accurately infer a solution's pH value from the voltage generated 
by a glass electrode, it must "know" the expected slope of the Nernst equation. Since the only 
variable in the Nernst equation beside the two ion concentration values [C\ and C-i) is temperature 
(T), a simple temperature measurement will provide the pH instrument the information it needs to 
function accurately. For this reason, many pH instruments are constructed to accept an RTD input 
for solution temperature sensing, and many pH probe assemblies have built-in RTD temperature 
sensors ready to sense solution temperature. 

The slope of a pH instrument is generally set by performing a two-point calibration using buffer 
solutions as the pH calibration standard. A buffer solution is a specially formulated solution that 
maintains a stable pH value even under conditions of slight contamination. For more information 
on pH buffer solutions, see section 11.8.5 on page 285. The pH probe assembly is inserted into a 
cup containing a buffer solution of known pH value, then the pH instrument is "standardized" to 



that pH value 13 . After standardizing at the first calibration point, the pH probe is removed from 
the buffer, rinsed, then placed into another cup containing a second buffer with a different pH value. 
After another stabilization period, the pH instrument is standardized to this second pH value. 

It only takes two points to define a line, so these two buffer measurements are all that is required 
by a pH instrument to define the linear transfer function relating probe voltage to solution pH: 

4.02 pH buffer 

1 0.06 pH buffer 

1 20 -60 

Probe voltage (mV) 

1 1 

80 240 300 

-165 mV 


Most modern pH instruments will display the calculated slope value after calibration. This value 
should (ideally) be 59.17 millivolts per pH unit at 25 degrees Celsius, but it will likely be a bit less 
than this. The voltage-generating ability of a glass electrode decays with age, so a low slope value 
may indicate a probe in need of replacement. 

Another informative feature of the voltage/pH transfer function graph is the location of the 
isopotential point: that point on the graph corresponding to voltage. In theory, this point should 
correspond to a pH value of 7.0 pH. However, if there exist stray potentials in the pH measurement 
circuit - for example, voltage differences caused by ion mobility problems in the porous junction of 
the reference electrode - this point will be shifted. A quick way to check the isopotential point of 
any calibrated pH instrument is to short the input terminals together (forcing Vi npu t to be equal to 
millivolts) and note the pH indication on the instrument's display. 

When calibrating a pH instrument, you should choose buffers that most closely "bracket" the 
expected range of pH measurement in the process. The most common buffer pH values are 4, 7, and 
10 (nominal). For example, if you expect to measure pH values in the process ranging between 7.5 
and 9, you should calibrate that pH instrument using 7 and 10 buffers. 

13 With all modern pH instruments being digital in design, this standardization process usually entails pressing a 
pushbutton on the faceplate of the instrument to "tell" it that the probe is stabilized in the buffer solution. 


16.5 Chromatography 

Imagine a major marathon race, where hundreds of runners gather in one place to compete. When 
the starting gun is fired, all the runners begin running the race, starting from the same location (the 
starting line) at the same time. As the race progresses, the faster runners distance themselves from 
the slower runners, resulting in a dispersion of runners along the race course over time. 

Now imagine a marathon race where certain runners share the exact same running speeds. 
Suppose a group of runners in this marathon all run at exactly 8 miles per hour (MPH), while 
another group of runners in the race run at exactly 6 miles per hour, and another group runs at 
exactly 5 miles per hour. What would happen to these three groups of runners over time, supposing 
they all begin the race at the same location and at the exact same time? 

As you can probably imagine, the runners within each speed group will stay with each other 
throughout the race, with the three groups becoming further spread apart over time. The first of 
these three groups to cross the finish line will be the 8 MPH runners, followed by the 6 MPH runners 
a bit later, and then followed by the 5 MPH runners after that. To an observer at the very start of 
the race, it would be difficult to tell exactly how many 6 MPH runners there were in the crowd, but 
to an observer at the finish line with a stop watch, it would be very easy to tell how many 6 MPH 
runners competed in the race (by counting how many runners crossed the finish line at the exact 
time corresponding to a speed of 6 MPH). 

Now imagine a mixture of chemicals in a fluid state traveling through a very small-diameter 
"capillary" tube filled with an inert, porous material such as sand. Some of those fluid molecules 
will find it easier to progress down the length of the tube than others, with similar molecules sharing 
similar propagation speeds. Thus, a small sample of that chemical mixture injected into such a 
capillary tube, and carried along the tube by a continuous flow of solvent (gas or liquid), will 
tend to separate into its constituent components over time just like the crowd of marathon runners 
separate over time according to running speed. A detector placed at the outlet of the capillary tube, 
configured to detect any chemical different from the solvent, will indicate the different components 
exiting the tube at different times. If the "running speed" of each chemical component is known 
from prior tests, this device may be used to identify the composition of the original chemical mix 
(and even how much of each component was present in the injected sample). 

This is the essence of chromatography: the technique of chemical separation by time-delayed 
travel down the length of a stationary medium (called a column). In chromatography, the chemical 
solution traveling down the column is called the mobile phase, while the solid and/or liquid substance 
residing within the column is called the stationary phase. Chromatography was first applied to 
chemical analysis by a Russian botanist named Tswett, who was interested in separating mixtures 
of plant pigments. The colorful bands left behind in the stationary phase by the separated pigments 
gave rise to the name "chromatography," which literally means "color writing." 

Modern chemists often apply chromatographic techniques in the laboratory to purify chemical 
samples, and/or to measure the concentrations of different chemical substances within mixtures. 
Some of these techniques are manual (such as in the case of thin-layer chromatography, where liquid 
solvents carry liquid chemical components along a flat plate covered with an inert coating such 
as alumina, and the positions of the chemical drops after time distinguishes one component from 
another). Other techniques are automated, with machines called chromatographs performing the 
timed analysis of chemical travel through tightly-packed tubular columns. 



Step 1 

Thin-layer chromatography 
Step 2 Step 3 






= Sample 

Solvent "wicks" 
up the plate 



= Component "A" 
= Component "B" 


= Component "A" 
= Component "B" 

As solvent wicks up the surface of the plate, it carries along with it all 
components of the sample spot. Each component travels at a different 
speed, separating the components along the plate over time. 

The simplest forms of chromatography reveal the chemical composition of the analyzed mixture 
as residue retained by the stationary phase. In the case of thin-layer chromatography, the different 
liquid components of the mobile phase remain embedded in the stationary phase at distinct locations 
after sufficient "developing" time. The same is true in paper-strip chromatography where a simple 
strip of filter paper serves as the stationary phase through which the mobile phase (liquid sample 
and solvent) travels: the different components of the sample remain in the paper as residue, their 
relative positions along the paper's length indicating their extent of travel during the test period. 
If the components have different colors, the result will be a stratified pattern of colors on the paper 
strip . 

Most chromatography techniques, however, allow the sample to completely wash through a 
packed column, relying on a detector at the end of the column to indicate when each component 
has exited the column. A simplified schematic of a process gas chromatograph (GC) shows how this 
type of analyzer functions: 

14 This effect is particularly striking when paper-strip chromatography is used to analyze the composition of ink. It 
is really quite amazing to see how many different colors are contained in plain "black" ink! 



Sample in 

Pressure regulator 


Sample valve 



Shutoff valve 

Carrier gas 

(for gas chromatographs only) 



Sample out 





The sample valve periodically injects a very precise quantity of sample into the entrance of the 
column tube and then shuts off to allow the constant-flow carrier gas to wash this sample through the 
length of the column tube. Each component of the sample travels through the column at different 
rates, exiting the column at different times. All the detector needs to do is be able to tell the 
difference between pure carrier gas and carrier gas mixed with anything else (components of the 
sample) . 

Several different detector designs exist for process gas chromatographs. The two most common 
are the flame ionization detector (FID) and the thermal conductivity detector (TCD). All detectors 
exploit some physical difference between the solutes (sample components dissolved within the carrier 
gas) and the carrier gas itself which acts as a gaseous solvent. 

Flame ionization detectors work on the principle of ions liberated in the combustion of the sample 
components. A permanent flame (usually fueled by hydrogen gas which produces negligible ions in 
combustion) serves to ionize any gas molecules exiting the chromatograph column that are not 
carrier gas. Common carrier gases used with FID sensors are helium and nitrogen. Gas molecules 
containing carbon easily ionize during combustion, which makes the FID sensor well-suited for GC 
analysis in the petrochemical industries, where hydrocarbon content analysis is the most common 
form of analytical measurement 15 . 

Thermal conductivity detectors work on the principle of heat transfer by convection (gas cooling) . 
Recall the dependence of a thermal mass flowmeter's calibration on the specific heat value of the gas 
being measured 16 . This dependence upon specific heat meant that we needed to know the specific 
heat value of the gas whose flow we intend to measure, or else the flowmeter's calibration would be 

15 In fact, FID sensors are sometimes referred to as carbon counters, since their response is almost directly 
proportional to the number of carbon atoms passing through the flame. 

16 See section 15.6, on page 526. The greater the specific heat value of a gas, the more heat energy it can carry away 
from a hot object through convection, all other factors being equal. 



in jeopardy. Here, in the context of chromatograph detectors, we exploit the impact specific heat 
value has on thermal convection, using this principle to detect compositional change for a constant- 
flow gas rate. The temperature change of a heated RTD or thermistor caused by exposure to a 
gas mixture with changing specific heat value indicates when a new sample component exits the 
chromatograph column. 

If we plot the response of the detector on a graph, we see a pattern of peaks, each one 
indicating the departure of a component "group" exiting the column. This graph is typically called 
a chromatogram: 




1 = First component to exit column 
5 = Last component to exit column 


Narrow peaks represent compact bunches of molecules all exiting the column at nearly the same 
time. Wide peaks represent more diffuse groupings of similar (or identical) molecules. In this 
chromatogram, you can see that components 4 and 5 are not clearly differentiated over time. Better 
separation may be achieved by altering the sample volume, carrier gas flow rate, type of carrier gas, 
column packing material, and/or column temperature. 

If the relative propagation speeds of each component is known in advance, the chromatogram 
peaks may be used to identify the presence (and quantities of) those components. The quantity 
of each component present in the original sample may be determined by applying the calculus 
technique of integration to each chromatogram peak, calculating the area underneath each curve. 
The vertical axis represents detector signal, which is proportional to component concentration 17 

17 Detector response also varies substantially with the type of substance being detected, and not just its 
concentration. A flame ionization detector (FID), for instance, yields different responses for a given mass flow rate of 
butane (C4H10) than it does for the same mass flow rate of methane (CH4), due to the differing carbon count per 
mass ratios of the two compounds. This means the same raw signal from an FID sensor generated by a concentration 
of butane versus a concentration of methane actually represents different concentrations of butane versus methane 



which is proportional to flow rate given a fixed carrier flow rate. This means the height of each peak 
represents mass flow rate of each component (W , in units of micrograms per minute, or some similar 
units). The horizontal axis represents time, so therefore the integral (sum of infinitesimal products) 
of the detector signal over the time interval for any specific peak (time t\ to t%) represents a mass 
quantity that has passed through the column. In simplified terms, a mass flow rate (micrograms per 
minute) multiplied by a time interval (minutes) equals mass in micrograms: 




m = Mass of sample component in micrograms 

W = Instantaneous mass flow rate of sample component in micrograms per minute 

t = Time in minutes {t\ and £2 ar e the interval times between which total mass is calculated) 

This mathematical relationship may be seen in graphical form by shading the area underneath 
the peak of a chromatogram: 




(Hg/min) £ 



Area accumulated under the curve 
represents the total mass of that 
component passed through the 
detector between times t, and t 2 

Time - 

Since process chromatographs have the ability to independently analyze the quantities of multiple 
components in a chemical sample, these instruments are inherently multi-variable. A single analog 

in the carrier. The inconsistent response of a chromatograph detector to different sampled components is not as 
troubling a problem as one might think, though. Since the chromatograph column does a good job separating each 
component from the other over time, we may program the computer to re-calibrate itself for each component at the 
specific time(s) each component is expected to exit the column. So long as we know in advance the characteristic 
detector response for each expected compound separated by the chromatograph, we may easily compensate for those 
variations in real time so that the chromatogram consistently and accurately represents component concentrations 
over the entire analysis cycle. 


output signal (e.g. 4-20 mA) would only be able to transmit information about the concentration 
of any one component (any one peak) in the chromatogram. This is perfectly adequate if only one 
component concentration is worth knowing about in the process 18 , but some form of multi-channel 
digital (or multiple analog outputs) transmission is necessary to make full use of a chromatograph's 

All modern chromatographs are "smart" instruments, containing one or more digital computers 
which execute the calculations necessary to derive precise measurements from chromatogram data. 
The computational power of modern chromatographs may be used to further analyze the process 
sample, beyond simple determinations of concentration or quantity. Examples of more abstract 
analyses include approximate octane value of gasoline (based on the relative concentrations of several 
components), or the heating value of natural gas (based on the relative concentrations of methane, 
ethane, propane, butane, carbon dioxide, helium, etc. in a sample of natural gas). The following 
photograph shows a gas chromatograph (GC) fulfilling precisely this purpose: the determination of 
heating value for natural gas 19 . 

18 It is not uncommon to find chromatographs used in processes to measure the concentration of a single chemical 
component, even though the device is capable of measuring the concentrations of multiple components in that process 
stream. In those cases, chromatography is (or was at the time of installation) the most practical analytical technique 
to use for quantitative detection of that substance. Why else use an inherently mult i- variable analyzer when you could 
have used a different, single- variable technology that was single- variable? By analogy, it is possible to use a Coriolis 
flowmeter to measure nothing but fluid density, even though such a device is fully capable of measuring fluid density 
and mass flow rate and temperature. 

19 Since the heat of combustion is well-known for various components of natural gas (methane, ethane, propane, etc.), 
all the chromatograph computer needs to do is multiply the different heat values by their respective concentrations 
in the gas flowstream, then average the total heat value per unit volume (or mass) of natural gas. 



This particular GC is used by a natural gas distribution company as part of its pricing system. 
The heating value of the natural gas is used as data to calculate the selling price of the natural gas 
(dollars per standard cubic foot), so that the customers pay only for the actual benefit of the gas 
(i.e. its ability to function as a fuel) and not just volumetric or mass quantity. 

Although the column cannot be seen in the photograph of the GC, several high-pressure steel 
"bottles" may be seen in the background holding carrier gas used to wash the natural gas sample 
through the column. A typical gas chromatograph column appears in the next photograph. It is 
nothing more than a stainless-steel tube packed with an inert, porous filling material: 



This particular GC column is 28 feet long, with an outside diameter of only 1/8 inch (the tube's 
inside diameter is even less than that). Column geometry and packing material vary greatly with 
application. The many choices intrinsic to column design are best left to specialists in the field of 
chromatography, not the average technician or even the average process engineer. 

Arguably, the most important component of a process gas chromatograph is the sample valve. 
Its purpose is to inject the exact same sample quantity into the column at the beginning of each 
cycle. A common form of sample valve uses a rotating element to switch port connections between 
the sample gas stream, carrier gas stream, and column: 



Position 2 




Fluid path \l[ R otor 


Three slots connect three pairs of ports together. When the rotary valve actuates, the port 
connections switch, redirecting gas flows. 

Connected to a sample stream, carrier stream, and column, the rotary sample valve operates in 
two different modes. The first mode is a "loading" position where the sample stream flows through 
a short length of tubing (called a sample loop) and exits to a waste discharge port, while the carrier 
gas flows through the column to wash the last sample through. The second mode is a "sampling" 
position where the volume of sample gas held in the sample loop tubing gets injected into the column 
by a flow of carrier gas behind it: 



Sample loop 

Loading position 

Sample in 

To waste -+ 

$ »-To column 

Carrier gas 

Sample loop 

Sampling position 

-) »-To column 

Carrier gas 

The purpose of the sample loop tube is to act as a holding reservoir for a fixed volume of sample 
gas. When the sample valve switches to the sample position, the carrier gas will flush the contents of 
the sample loop into the front of the column. This valve configuration guarantees that the injected 
sample volume does not vary with inevitable variations in sample valve actuation time. The sample 
valve need only remain in the "sampling" position long enough to completely flush the sample loop 
tube, and the proper volume of injected sample gas is guaranteed. 

While in the loading position, the stream of gas sampled from the process continuously fills the 
sample loop and then exits to a waste port. This may seem unnecessary but it is in fact essential for 
practical sampling operation. The volume of process gas injected into the chromatograph column 
during each cycle is so small (typically measured in units of microliters!) that a continuous flow of 
sample gas to waste is necessary to purge the impulse line connecting the analyzer to the process, 
which in turn is necessary for the analyzer to obtain analyses of current conditions. If it were not 
for the continuous flow of sample to waste, it would take a very long time for a sample of process 



gas to make its way through the long impulse tube to the analyzer to be sampled! 

<, Process pipe 

Block valve 

Impulse line 


(to vent, flare, or 
other safe location) 


Sample conditioning 

(cooling, heating, filtering) 


-> Signal output(s) 

Even with continuous flow in the impulse line, process chromatographs exhibit substantial dead 
time in their analyses for the simple reason of having to wait for the next sample to progress through 
the entire length of the column. It is the basic nature of a chromatograph to separate components 
of a chemical stream over time, and so a certain amount of dead time will be inevitable. However, 
dead time in any measuring instrument is an undesirable quality. Dead time in a feedback control 
loop is especially bad, as it greatly increases the chances of instability. 

One way to reduce the dead time of a chromatograph is to alter some of its operating parameters 
during the analysis cycle in such a way that it speeds up the progress of the mobile phase during 
periods of time where slowness of elution is not as important for fine separation of components. The 
flow rate of the mobile phase may be altered, the temperature of the column may be ramped up or 
down, and even different columns may be switched into the mobile phase stream. In chromatography, 
we refer to this on-line alteration of parameters as programming. Temperature programming is an 
especially popular feature of process gas chromatographs, due to the direct effect temperature has 
on the viscosity of a flowing gas 20 . Carefully altering the operating temperature of a GC column 
while a sample washes through it is an excellent way to optimize the separation and time delay 
properties of a column, effectively realizing the high separation properties of a long column with the 
reduced dead time of a much shorter column. 

20 Whereas most liquids decrease in viscosity as temperature rises, gases increase in viscosity as they get hotter. 
Since the flow regime through a chromatograph column is most definitely laminar and not turbulent, viscosity has a 
great effect on flow rate. 



Boylestad, Robert L., Introductory Circuit Analysis, 9th Edition, Prentice Hall, Upper Saddle River, 
New Jersey, 2000. 

Fribance, Austin E., Industrial Instrumentation Fundamentals, McGraw-Hill Book Company, New 
York, NY, 1962. 

Kohlmann, Frederick J., What Is pH, And How Is It Measured?, Hach Company, 2003. 

Lavigne, John R., Instrumentation Applications for the Pulp and Paper Industry, Miller Freeman 
Publications, Foxboro, MA, 1979. 

Liptak, Bela C, Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Novak, Joe, What Is Conductivity, And How Is It Measured?, Hach Company, 2003. 

Sadar, Michael J., Turbidity Science, Technical Information Series - Booklet No. 11, Hach Company, 

Scott, Raymond P.W., Gas Chromatography, Library4Science, LLC, 2003. 

Scott, Raymond P.W., Gas Chromatography Detectors, Library4Science, LLC, 2003. 

Scott, Raymond P.W., Liquid Chromatography, Library4Science, LLC, 2003. 

Scott, Raymond P.W., Liquid Chromatography Detectors, Library4Science, LLC, 2003. 

Scott, Raymond P.W., Principles and Practice of Chromatography, Library4Science, LLC, 2003. 

Sherman, R.E.; Rhodes, L.J., Analytical Instrumentation: practical guides for measurement and 
control, ISA, Research Triangle Park, NC, 1996. 

Shinskey, Francis C, pH and pION Control in Process and Waste Streams, John Wiley & Sons, New 
York, NY, 1973. 

Theory and Practice of pH Measurement, PN 44-6033, Rosemount Analytical, 1999. 

Chapter 17 

Signal characterization 

Mathematics is full of complementary principles and symmetry. Perhaps nowhere is this more 
evident than with inverse functions: functions that "un-do" one another when put together. A few 
examples of inverse functions are shown in the following table: 


r 1 ^) 











Inverse functions are vital to master if one hopes to be able to manipulate algebraic (literal) 
expressions. For example, to solve for time (t) in this exponential formula, you must know that the 
natural logarithm function directly "un-does" the exponential e x . This is the only way to "unravel" 
the equation and get t isolated by itself on one side of the equals sign: 

V = 12e~* 

Divide both sides by 12 


Take the natural logarithm of both sides 


The natural logarithm "cancels out" the exponential 



Multiply both sides by negative one 


In industry there exist a great many practical problems where inverse functions play a similar 
role. Just as inverse functions are useful for manipulating literal expressions in algebra, they are also 
useful in inferring measurements of things we cannot directly measure. Many continuous industrial 
measurements are inferential in nature, meaning that we actually measure some other variable in 
order to quantify the variable of interest. More often than not, the relationship between the primary 
variable and the inferred variable is nonlinear, necessitating some form of mathematical processing 
to complete the inferential measurement. 

Take for instance the problem of measuring fluid flow through a pipe. To the layperson, this 
may seem to be a trivial problem. However there is no practical way to directly and continuously 
measure the flow rate of a fluid, especially when we cannot allow the fluid in question to become 
exposed to the atmosphere (e.g. when the liquid or gas in question is toxic, flammable, under high 
pressure, or any combination thereof). 

One standard way to measure the flow rate of a fluid through a pipe is to intentionally place a 
restriction in the path of the fluid, and measure the pressure drop across that restriction. The most 
common form of intentional restriction used for this purpose is a thin plate of metal with a hole 
precisely machined in the center, called an orifice plate. A side view of the orifice plate assembly 
and pressure-measuring instrument looks like this: 





Direction of flow 

This approach should make intuitive sense: the faster the flow rate of the fluid, the greater the 
pressure difference developed across the orifice. The actual physics of this process has to do with 
energy exchanging between potential and kinetic forms, but that is incidental to this discussion. 
The mathematically interesting characteristic of this flow measurement technique is its nonlinearity. 
Pressure does not rise linearly with flow rate; rather, it increases with the square of the flow rate: 

Diff. pressure 


Flow (Q) 

To write this as a proportionality, we relate flow rate (Q) to pressure (P) as follows (the constant 
k accounts for unit conversions and the geometries of the orifice plate and pipe) : 

P = kQ 2 

This is a practical problem for us because our intent is to use pressure measurement (P) as an 
indirect (inferred) indication of flow rate (Q). If the two variables are not directly related to one 



another, we will not be able to regard one as being directly representative of the other. To make 
this problem more clear to see, imagine a pressure gauge connected across the restriction, with the 
face of the gauge labeled in percent: 

Face of pressure gauge, calibrated to 
read in percent of full flow rate 

Consider a pressure gauge such as the one shown above, registering 20 percent on a linear scale 
at some amount of flow through the pipe. What will happen if the flow rate through that pipe 
suddenly doubles? An operator or technician looking at the gauge ought to see a new reading of 40 
percent, if indeed the gauge is supposed to indicate flow rate. However, this will not happen. Since 
the pressure dropped across the orifice in the pipe increases with the square of flow rate, a doubling 
of flow rate will actually cause the pressure gauge reading to quadruple! In other words, it will go 
from reading 20% to reading 80%, which is definitely not an accurate indication of the flow increase. 

A couple of simple solutions exist for addressing this problem. One is to re-label the pressure 
gauge with a "square root" scale. Examine this photograph of a 3-15 PSI receiver gauge having 
both linear and square-root scales: 


Now, a doubling of fluid flow rate still results in a quadrupling of needle motion, but due to the 
nonlinear scale this translates into a simple doubling of indicated flow, which is precisely what we 
need for this to function as an accurate flow indicator. 

If the differential pressure instrument outputs a 4-20 mA analog electronic signal instead of a 
3-15 PSI pneumatic signal, we may apply the same "nonlinear scale" treatment to any current meter 
and achieve the same result: 



Another simple solution is to use a nonlinear manometer, with a curved viewing tube 1 : 


Curved manometer 


The scale positioned alongside the curved viewing tube will be linear, with equal spacings between 
division marks along its entire length. The vertical height of the liquid column translates pressure 
into varying degrees of movement along the axis of the tube by the tube's curvature. Literally, any 
inverse function desired may be "encoded" into this manometer by fashioning the viewing tube into 
the desired (custom) shape without any need to print a nonlinear scale. 

1 This solution works best for measuring the flow rate of gases, not liquids, since the manometer obviously must use 
a liquid of its own to indicate pressure, and mixing or other interference between the process liquid and the manometer 
liquid could be problematic. 


Shown here is a photograph of an actual curved-tube manometer. This particular specimen does 
not have a scale reading in units of flow, but it certainly could if it had the correct curve for a 
square-root characterization: 

A more sophisticated solution to the "square root problem" is to use a computer to manipulate 
the signal coming from the differential pressure instrument so that the characterized signal becomes a 
direct, linear representation of flow. In other words, the computer square-roots the pressure sensor's 
signal in order that the final signal becomes a direct representation of fluid flow rate: 




pressure _. . . Indicating 

instrument Charactenzer gau ge 

Direction of flow 

Both solutions achieve their goal by mathematically "un-doing" the nonlinear (square) function 
intrinsic to the physics of the orifice plate with a complementary (inverse) function. This intentional 
compounding of inverse functions is sometimes called linearization, because it has the overall effect 
of making the output of the instrument system a direct proportion of the input: 

Output = fc(Input) 

Fluid flow rate measurement in pipes is not the only application where we find nonlinearities 
complicating the task of measurement. Several other applications exhibit similar challenges: 

• Liquid flow measurement in open channels (over weirs) 

• Liquid level measurement in non-cylindrical vessels 

• Temperature measurement by radiated energy 

• Chemical composition measurement 

The following sections will describe the mathematics behind each of these measurement 



17.1 Flow measurement in open channels 

Measuring the flow rate of liquid through an open channel is not unlike measuring the flow rate of a 
liquid through a closed pipe: one of the more common methods for doing so is to place a restriction 
in the path of the liquid flow and then measure the "pressure" dropped across that restriction. The 
easiest way to do this is to install a low "dam" in the middle of the channel, then measure the height 
of the liquid upstream of the dam as a way to infer flow rate. This dam is technically referred to as 
a weir, and three styles of weir are commonly used: 

Different styles of weirs for measuring open-channel liquid flow 




Another type of open-channel restriction used to measure liquid flow is called a flume. An 
illustration of a Parshall flume is shown here: 

Weirs and flumes may be thought of being somewhat like "orifice plates" and "venturi tubes," 
respectively, for open-channel liquid flow. Like an orifice plate, a weir or a flume generates a 
differential pressure that varies with the flow rate through it. However, this is where the similarities 
end. Exposing the fluid stream to atmospheric pressure means the differential pressure caused 
by the flow rate manifests itself as a difference in liquid height at different points in the channel. 
Thus, weirs and flumes allow the indirect measurement of liquid flow by sensing liquid height. An 
interesting feature of weirs and flumes is that although they are nonlinear primary sensing elements, 
their nonlinearity is quite different from that of an orifice. 

Note the following transfer functions for different weirs and flumes, relating the rate of liquid 
flow through the device (Q) to the level of liquid rise upstream of the device (called "head", or H): 


Q = 2.48 [ tan — ) H% V-notch weir 

Q = 3.367 LH * Cippoletti weir 
Q = 0.992 J ff 1547 3-inch wide throat Parshall flume 

Q = 3.07H 153 9-inch wide throat Parshall flume 


Q = Volumetric flow rate (cubic feet per second - CFS) 
L = Width of notch crest or throat width (feet) 
= V-notch angle (degrees) 
H = Head (feet) 

It is important to note these functions provide answers for flow rate (Q) with head (H) being 
the independent variable. In other words, they will tell us how much liquid is flowing given a certain 
head. In the course of calibrating the head-measuring instruments that infer flow rate, however, it 
is important to know the inverse transfer function: how much head there will be for any given value 
of flow. Here, algebraic manipulation becomes important to the technician. For example, here is 
the solution for H in the function for a Cippoletti weir: 

Q = 3.367LHi 
Dividing both sides of the equation by 3.367 and L: 





Taking the - root of both sides: 




This in itself may be problematic, as some calculators do not have an ?/y function. In cases such 
as this, it is helpful to remember that a root is nothing more than an inverse power. Therefore, we 

| power instead of a | 

could re-write the final form of the equation using a I power instead of a | root 

Q , =H 


17.2 Liquid volume measurement 

When businesses use large vessels to store liquids, it is useful to know how much liquid is stored in 
each vessel. A variety of technologies exist to measure stored liquid. Hydrostatic pressure, radar, 
ultrasonic, and tape-and-float are just a few of the more common technologies: 


Pressure sensor infers 
liquid level by measuring 
static pressure developed 
by the liquid "head" 

Radar or Ultrasonic 



Radio or sound waves 
bounced off the liquid surface 
determine how far away the 

liquid is from the sensor 

Float riding on liquid 
surface moves a metal 
cable or "tape," which 
directly registers level 

These liquid measuring technologies share a common trait: they measure the quantity of liquid 
in the vessel by measuring liquid height. If the vessel in question has a constant cross-sectional area 
throughout its working height (e.g. a vertical cylinder), then liquid height will directly correspond 
to liquid volume. However, if the vessel in question does not have a constant cross-sectional area 
throughout its height, the relationship between liquid height and liquid volume will not be linear. 

For example, there is a world of difference between the height/volume functions for a vertical 
cylinder versus a horizontal cylinder: 












The volume function for a vertical cylinder is a simple matter of geometry - height (h) multiplied 
by the cylinder's cross-sectional area (irr 2 ): 


irr h 



Calculating the volume of a horizontal cylinder as a function of liquid height (h) is a far more 
complicated matter, because the cross-sectional area is also a function of height. For this, we need 
to apply calculus. 

First, we begin with the mathematical definition of a circle, then graphically represent a partial 
area of that circle as a series of very thin rectangles: 

In this sketch, I show the circle "filling" from left to right rather than from bottom to top. I have 
done this strictly out of mathematical convention, where the x (horizontal) axis is the independent 
variable. No matter how the circle gets filled, the relationship of area (A) to fill distance (h) will be 
the same. 

If x 2 + y 2 = r 2 (the mathematical definition of a circle) , then the area of each rectangular "slice" 
comprising the accumulated area between — r and h — r is equal to 2y dx. In other words, the total 
accumulated area between — r and h — r is: 

h — r 

2y dx 

Now, writing y in terms of r and x (y = \/r 2 — x 2 ) and moving the constant "2" outside the 

\/r 2 — x 2 dx 

Consulting a table of integrals, we find this solution for the general form: 


\/a 2 — u 2 du = — \/a 2 — u 2 -\ sin ( — ) + C 

2 2 \aJ 



Applying this solution to our particular integral 

- v r 2 

h — r 



- v / r 2 -{h-r) 2 + 

r , (h - r) 

— sin 

2 r 

- V ?" 2 — (—r) 2 H sin 


-^-^r 2 - th 2 - 2hr + r 2 ) + — sin" 1 ^LJj 
2 2 r 


r — n 

'(^1^-V + 

r 2 _, 0- 

— sin 

2 r 

K.2\ 1 

(ft, - r) \J2hr - h 2 + r 2 sin -1 ^ ^ + — 

Knowing that the stored liquid volume in the horizontal tank will be this area multiplied by the 
constant length (L) of the tank, our formula for volume is as follows: 


(h - r)V2hr - h 2 + r 2 sin" 1 ^ ^ 


As you can see, the result is far from simple. Any instrumentation system tasked with the 
inference of stored liquid volume by measurement of liquid height in a horizontal cylinder must 
somehow apply this formula on a continuous basis. This is a prime example of how digital computer 
technology is essential to certain continuous measurement applications! 

Spherical vessels, such as those used to store liquefied natural gas (LNG) and butane, present a 
similar challenge. The height/volume function is nonlinear because the cross-sectional area of the 
vessel changes with height. 

Calculus provides a way for us to derive an equation solving for stored volume (V) with height (/i) 
as the independent variable. We begin in a similar manner to the last problem with the mathematical 
definition of a circle, except now we consider the filling of a sphere with a series of thin, circular 




If x 2 + y 2 = r 2 (the mathematical definition of a circle) , then the volume of each circular disk 
comprising the accumulated volume between — r and h — r is equal to iry 2 dx. In other words, the 
total accumulated area between — r and h — r is: 



Try dx 

Now, writing y in terms of r and x (y = \/r 2 — x 2 ) and moving the constant 7r outside the 

V = 7T 

h — r 

I v r 2 — x 2 ) 


Immediately we see how the square and the square-root cancel one another, leaving us with a 
fairly simple integrand: 

V = 7T 


r — x dx 

We may write this as the difference of two integrals: 

I f^ 2 ^ I f^ 2 

V = I 7r / r dx I — I 7r / x dx 

Since r is a constant, the left-hand integral is simply 7rr 2 a:. The right-hand integral is solvable 
by the power rule: 



1/ 2 r ih— r 

V = irr [x\_ r 

V = Trr 2 \{h - r) - (-r) 

(h-r) 3 (-r) 

7T 7T 


V = nr 2 [h - r + r] - - [(h - rf - (-r) 3 ] 

V = Trhr 2 - - [(h 3 - 2h 2 r + hr 2 - h 2 r + 2hr 2 - r 3 ) + 

V = irhr 2 \h 3 - Zh 2 r + 3hr 

V = irhr h 7r/i r — irhr 



irh 3 

+ nh r 

V = irh 2 r 

irh 3 

V = irh 2 (r- -) 

This function will "un-do" the inherent height /volume nonlinearity of a spherical vessel, allowing 
a height measurement to translate directly into a volume measurement. A "characterizing" function 
such as this is typically executed in a digital computer connected to the level sensor, or sometimes 
in a computer chip within the sensor device itself. 

An interesting alternative to a formal equation for linearizing the level measurement signal is to 
use something called a multi-segment characterizer function, also implemented in a digital computer. 
This is an example of what mathematicians call a piecewise function: a function made up of line 
segments. Multi-segment characterizer functions may be programmed to emulate virtually any 
continuous function, with reasonable accuracy: 



Continuous characterizing function 

Piecewise characterizing function 







The computer correlates the input signal (height measurement, h) to a point on this piecewise 
function, linearly interpolating between the nearest pair of programmed coordinate points. The 
number of points available for multi-point characterizers varies between ten and one hundred 2 
depending on the desired accuracy and the available computing power. 

Although true fans of math might blanch at the idea of approximating an inverse function for level 
measurement using a piecewise approach rather than simply implementing the correct continuous 
function, the multi-point characterizer technique does have certain practical advantages. For one, 
it is readily adaptable to any shape of vessel, no matter how strange. Take for instance this vessel, 
made of separate cylindrical sections welded together: 




Here, the vessel's very own height/volume function is fundamentally piecewise, and so nothing 
but a piecewise characterizing function could possibly linearize the level measurement into a volume 

2 There is no theoretical limit to the number of points in a digital computer's characterizer function given sufficient 
processing power and memory. There is, however, a limit to the patience of the human programmer who must encode 
all the necessary x, y data points defining this function. Most of the piecewise characterizing functions I have seen 
available in digital instrumentation systems provide 10 to 20 (x, y) coordinate points to define the function. Fewer 
than 10 coordinate points risks excessive interpolation errors, and more than 20 would just be tedious to set up. 



Consider also the case of a spherical vessel with odd-shaped objects welded to the vessel walls, 
and/or inserted into the vessel's interior: 

The volumetric space occupied by these structures will introduce all kinds of discontinuities into 
the transfer function, and so once again we have a case where a continuous characterizing function 
cannot properly linearize the level signal into a volume measurement. Here, only a piecewise function 
will suffice. 

To best generate the coordinate points for a proper multi-point characterizer function, one must 
collect data on the storage vessel in the form of a strapping table. This entails emptying the vessel 
completely, then filling it with measured quantities of liquid, one sample at a time, and taking level 

Introduced liquid volume 

Measured liquid level 

150 gallons 

2.46 feet 

300 gallons 

4.72 feet 

450 gallons 

5.8 feet 

600 gallons 

(etc., etc.) 

750 gallons 

(etc., etc.) 

Each of these paired numbers would constitute the coordinates to be programmed into the 
characterizer function computer by the instrument technician or engineer: 


600 -i 



(gallons) 300 








1 2 3 4 5 6 
h (feet) 



Many "smart" level transmitter instruments have enough computational power to perform the 
level-to-volume characterization directly, so as to transmit a signal corresponding directly to liquid 
volume rather than just liquid level. This eliminates the need for an external "level computer" 
to perform the necessary characterization. The following screenshot was taken from a personal 
computer running configuration software for a radar level transmitter 3 , showing the strapping table 
data point fields where a technician or engineer would program the vessel's level-versus-volume 
piecewise function: 

W Configure/Setup of CTLR-01C04CH0Z [3300 Rev. 2] 

File Actions Help 

d ^ISJ^J 

*Q| *?| 

Basic Setup J Setup i Volume Ij LCD ] Analog Output 1 Advanced I Version I 

Tank Type 

|Ver Cjilindr T] Tank Diameter j 39.37 
Tank Height | 196.85 

Current Measurement — 
Level ! 813 in 

Volume I 47 gal ; 


ppmg Table 

Entries used 





Max entries 


| 0.00 in 



| 0.00 in 


| 0.00 in 



| 0.00 in 


| 0.00 in 

| 0.00 in 


| 0.00 in 

| 0.00 in 


| 0.00 in 

| 0.00 in 


Time (Current 



| Cancel Apply Help 

|Device Last Synchronised: 8/18/2008 2:35:12 PM 

This configuration window actually shows more than just a strapping table. It also shows the 
option of calculating volume for different vessel shapes (vertical cylinder is the option selected here) 
including horizontal cylinder and sphere. In order to use the strapping table option, the user would 
have to select "Strapping Table" from the list of Tank Types. Otherwise, the level transmitter's 
computer will attempt to calculate volume from an ideal tank shape. 

3 The configuration software is Emerson's AMS, running on an engineering workstation in a DeltaV control system 
network. The radar level transmitter is a Rosemount model 3301 (guided-wave) unit. 


17.3 Radiative temperature measurement 

Temperature measurement devices may be classified into two broad types: contact and non- 
contact. Contact-type temperature sensors detect temperature by directly touching the material 
to be measured, and there are several varieties in this category. Non-contact temperature sensors 
work by detecting light emitted by hot objects. 

Energy radiated in the form of electromagnetic waves (photons, or light) relates to object 
temperature by an equation known as the Stefan-Boltzmann equation, which tells us the rate of 
heat lost by radiant emission from a hot object is proportional to the fourth power of its absolute 

P = eaAT 4 


P = Radiated energy power (watts) 

e = Emissivity factor (unitless) 

a = Stefan-Boltzmann constant (5.67 x 10~ 8 W / m 2 • K 4 ) 

A = Surface area (square meters) 

T = Absolute temperature (Kelvin) 

Solving for temperature (T) involves the use of the fourth root, to "un-do" the fourth power 
function inherent to the original function: 



Any optical temperature sensor measuring the emitted power (P) must "characterize" the power 
measurement using the above equation to arrive at an inferred temperature. This characterization 
is typically performed inside the temperature sensor by a microcomputer. 


17 A Analytical measurements 

A great many chemical composition measurements may be made indirectly by means of electricity, 
if those measurements are related to the concentration of ions (electrically charged molecules) . Such 
measurements include: 

• pH of an aqueous solution 

• Oxygen concentration in air 

• Ammonia concentration in air 

• Lead concentration in water 

The basic principle works like this: two different chemical samples are placed in close proximity 
to each other, separated only by an ion-selective membrane able to pass the ion of interest. As the 
ion activity attempts to reach equilibrium through the membrane, an electrical voltage is produced 
across that membrane. If we measure the voltage produced, we can infer the relative activity of the 
ions on either side of the membrane. 

Not surprisingly, the function relating ion activity to the voltage generated is nonlinear. The 
standard equation describing the relationship between ionic activity on both sides of the membrane 
and the voltage produced is called the Nernst equation: 

nF \a 2 


V = Electrical voltage produced across membrane due to ion exchange (volts) 

R = Universal gas constant (8.315 J/mol-K) 

T = Absolute temperature (Kelvin) 

n = Number of electrons transferred per ion exchanged (unitless) 

F = Faraday constant (96,485 coulombs per mole) 

Oi = Activity of ion in measured sample 

02 = Activity of ion in reference sample (on other side of membrane) 

A practical application for this technology is in the measurement of oxygen concentration in 
the flue gas of a large industrial burner, such as what might be used to heat up water to generate 
steam. The measurement of oxygen concentration in the exhaust of a combustion heater (or boiler) 
is very important both for maximizing fuel efficiency and for minimizing pollution (specifically, 
the production of NO^ molecules). Ideally, a burner's exhaust gas will contain no oxygen, having 
consumed it all in the process of combustion with a perfect stoichiometric mix of fuel and air. In 
practice, the exhaust gas of an efficiently-controlled burner will be somewhere near 2%, as opposed 
to the normal 21% of ambient air. 

One way to measure the oxygen content of hot exhaust is to use a high-temperature zirconium 
oxide detector. This detector is made of a "sandwich" of platinum electrodes on either side of a 
solid zirconium oxide electrolyte. One side of this electrochemical cell is exposed to the exhaust gas 
(process), while the other side is exposed to heated air which serves as a reference: 




Process gas 
(inside furnace) 

Platinum electrode 






O ' 

Reference gas (air) 

(ambient atmosphere) 

Platinum electrode 

The electrical voltage generated by this "sandwich" of zirconium and platinum is sent to an 
electronic amplifier circuit, and then to a microcomputer which applies an inverse function to the 
measured voltage in order to arrive at an inferred measurement for oxygen concentration. This 
type of chemical analysis is called potentiometric, since it measures ( "metric" ) based on an electrical 
voltage ("potential"). 

The Nernst equation is an interesting one to unravel, to solve for ion activity in the sample (a\) 
given voltage (V): 

RT fa x 

V=— In -1 

nr \a% 

Multiplying both sides by nF: 

nFV = RTln 

Dividing both sides by RT: 

nFV =l JaA 

RT \a 2 J 

Applying the rule that the difference of logs is equal to the log of the quotient: 



mcti — lno2 


Adding In 02 to both sides: 


I1102 = mai 

Making both sides of the equation a power of e: 

nFV 1 1 i 

e T5T^+ lna 2 _ g lna l 

Canceling the natural log and exponential functions on the right-hand side: 

nFV 1 i 

In most cases, the ionic activity of ai will be relatively constant, and so In 02 will be relatively 
constant as well. With this in mind, we may simplify the equation further, using k as our constant 

Substituting k for In a^'- 

e RT ^ = ai 

Applying the rule that the sum of exponents is the product of powers: 

e K e RT = ai 
If k is constant, then e will be constant as well (calling the new constant C): 


Ce RT = a 1 

Analytical instruments based on potentiometry must evaluate this inverse function to "undo" the 
Nernst equation to arrive at an inferred measurement of ion activity in the sample given the small 
voltage produced by the sensing membrane. These instruments typically have temperature sensors 
as well built in to the sensing membrane assembly, since it is apparent that temperature (T) also 
plays a role in the generation of this voltage. Once again, this mathematical function is typically 
evaluated in a microprocessor. 


Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Fourth Edition, CRC Press, New York, NY, 2003. 

Stewart, James, Calculus: concepts and contexts, 2nd Edition, Brooks/Cole, Pacific Grove, CA, 

Chapter 18 

Continuous feedback control 



18.1 Basic feedback control principles 

Instrumentation is the science of automated measurement and control. Applications of this science 
abound in modern research, industry, and everyday living. From automobile engine control systems 
to home thermostats to aircraft autopilots to the manufacture of pharmaceutical drugs, automation 
surrounds us. This section explains some of the fundamental principles of automatic process control. 
Before we begin our discussion on process control, we must define a few key terms. First, we 
have what is known as the process. This is the physical system we wish to monitor and control. For 
the sake of illustration, consider a heat exchanger that uses high-temperature steam to transfer heat 
to a lower-temperature liquid. Heat exchangers are used frequently in the chemical industries to 
maintain the necessary temperature of a chemical solution, so that the desired blending, separation, 
or reactions can occur. A very common design of heat exchanger is the "shell-and-tube" style, where 
a metal shell serves as a conduit for the chemical solution to flow through, while a network of smaller 
tubes runs through the heating space, carrying steam or some other heating medium. The hotter 
steam flowing through the tubes transfers heat energy to the cooler process fluid surrounding the 
tubes, inside the shell of the heat exchanger: 

heat exchanger 

Steam in 

Cool process / \\ \ Warm process 

fluid in 7 ■ V\ i 7 fluid out 

Steam out 

In this case, the process is the entire heating system, consisting of the fluid we wish to heat, 
the heat exchanger, and the steam delivering the required heat energy. In order to maintain steady 
control of the process fluid's exiting temperature, we must find a way to measure it and represent 
that measurement in signal form so that it may be interpreted by other instruments taking some 


form of control action. In instrumentation terms, the measuring device is known as a transmitter, 
because it transmits the process measurement in the form of a signal. Transmitters are represented 
in process diagrams by small circles with identifying letters inside, in this case, "TT," which stands 
for Temperature Transmitter: 

Steam in 

"Process Variable" (PV) 

Steam out 

The signal coming from the transmitter (shown in the illustration by the dashed line), 
representing the heated fluid's exiting temperature, is called the process variable. Like a variable 
in a mathematical equation that represents some story-problem quantity, this signal represents the 
measured quantity we wish to control in the process. 

In order to exert control over the process variable, we must have some way of altering fluid flow 
through the heat exchanger, either of the process fluid, the steam, or both. Generally, it makes 
more sense to alter the flow of the heating medium (the steam) , and let the process fluid flow rate 
be dictated by the demands of the larger process. If this heat exchanger were part of an oil refinery 
unit, for example, it would be far better to throttle steam flow to control oil temperature rather 
than to throttle the oil flow itself, since altering the oil's flow will undoubtedly affect other processes 
upstream and downstream of the exchanger. Ideally, the exchanger will act as a device that provides 
even, consistent temperature oil out, for any given temperature and flow-rate of oil in. 

One convenient way to throttle steam flow into the heat exchanger is to use a control valve 
(labeled "TV" because it is a Temperature Valve). In general terms, a control valve is known as a 
final control element. Other types of final control elements exist (servo motors, variable-flow pumps, 
and other mechanical devices used to vary some physical quantity at will), but valves are the most 



common, and probably the simplest to understand. With a final control element in place, the steam 
flow becomes known as the manipulated variable, because it is the quantity we will manipulate in 
order to gain control over the process variable: 

Steam in 

Control signal 

Steam out 

Valves come in a wide variety of sizes and styles. Some valves are hand-operated: that is, they 
have a "wheel" or other form of manual control that may be moved to "pinch off" or "open up" 
the flow passage through the pipe. Other valves come equipped with signal receivers and positioner 
devices, which move the valve mechanism to various positions at the command of a signal (usually 
an electrical signal, like the type output by transmitter instruments). This feature allows for remote 
control, so that a human operator or computer device may exert control over the manipulated 
variable from a distance. 

This brings us to the final, and most critical, component of the heat exchanger temperature 
control system: the controller. This is a device designed to interpret the transmitter's process 
variable signal and decide how far open the control valve needs to be in order to maintain that 
process variable at the desired value. 



Steam out 

Here, the circle with the letters "TC" in the center represents the controller. Those letters 
stand for Temperature Controller, since the process variable being controlled is the process fluid's 
temperature. Usually, the controller consists of a computer making automatic decisions to open and 
close the valve as necessary to stabilize the process variable at some predetermined setpoint. 

Note that the controller's circle has a solid line going through the center of it, while the 
transmitter and control valve circles are open. An open circle represents a field-mounted device 
according to the ISA standard for instrumentation symbols, and a single solid line through the 
middle of a circle tells us the device is located on the front of a control panel in a main control room 
location. So, even though the diagram might appear as though these three instruments are located 
close to one another, they in fact may be quite far apart. Both the transmitter and the valve must 
be located near the heat exchanger (out in the "field" area rather than inside a building), but the 
controller may be located a long distance away where human operators can adjust the setpoint from 
inside a safe and secure control room. 

These elements comprise the essentials of a feedback control system: the process (the system 
to be controlled), the process variable (the specific quantity to be measured and controlled), the 
transmitter (the device used to measure the process variable and output a corresponding signal), 
the controller (the device that decides what to do to bring the process variable as close to setpoint as 
possible), the final control element (the device that directly exerts control over the process), and the 



manipulated variable (the quantity to be directly altered to effect control over the process variable). 

Feedback control may be viewed as a sort of information "loop," from the transmitter (measuring 
the process variable), to the controller, to the final control element, and through the process itself, 
back to the transmitter. Ideally, a process control "loop" not only holds the process variable at a 
steady level (the setpoint), but also maintains control over the process variable given changes in 
setpoint, and even changes in other variables of the process: 






Final control 


The Process 

For example, if we were to raise the temperature setpoint in the heat exchanger process, the 
controller would automatically call for more steam flow by opening the control valve, thus introducing 
more heat energy into the process, thus raising the temperature to the new setpoint level. If 
the process fluid flow rate (an uncontrolled, or wild variable) were to suddenly increase, the heat 
exchanger outlet temperature would fall due to the physics of heat transfer, but once this drop was 
detected by the transmitter and reported to the controller, the controller would automatically call 
for additional steam flow to compensate for the temperature drop, thus bringing the process variable 
back in agreement with the setpoint. Ideally, a well-designed and well-tuned control loop will sense 
and compensate for any change in the process or in the setpoint, the end result being a process 
variable value that always holds steady at the setpoint value. 

Many types of processes lend themselves to feedback control. Consider an aircraft autopilot 
system, keeping an airplane on a steady course heading: reading the plane's heading (process 
variable) from an electronic compass and using the rudder as a final control element to change 
the plane's "yaw." An automobile's "cruise control" is another example of a feedback control 
system, with the process variable being the car's velocity, and the final control element being 
the engine's throttle. Steam boilers with automatic pressure controls, electrical generators with 



automatic voltage and frequency controls, and water pumping systems with automatic flow controls 
are further examples of how feedback may be used to maintain control over certain process variables. 

Modern technology makes it possible to control nearly anything that may be measured in an 
industrial process. This extends beyond the pale of simple pressure, level, temperature, and flow 
variables to include even certain chemical properties. 

In municipal water and wastewater treatment systems, numerous chemical quantities must be 
measured and controlled automatically to ensure maximum health and minimum environmental 
impact. Take for instance the chlorination of treated wastewater, before it leaves the wastewater 
treatment facility into a large body of water such as a river, bay, or ocean. Chlorine is added to the 
water to kill any residual bacteria so that they do not consume oxygen in the body of water they 
are released to. Too little chlorine added, and not enough bacteria are killed, resulting in a high 
biological oxygen demand or BOD in the water which will asphyxiate the fish swimming in it. Too 
much chlorine added, and the chlorine itself poses a hazard to marine life. Thus, the chlorine content 
must be carefully controlled at a particular setpoint, and the control system must take aggressive 
action if the dissolved chlorine concentration strays too low or too high: 

Chlorine supply 


4-20 mA 


4-20 mA i 
measurement i 

/^\ci 2 

Analytical / ._ | 
l Al I 

transmitter ' 



-> Effluent 

Now that we have seen the basic elements of a feedback control system, we will concentrate on 
the algorithms used in the controller to maintain a process variable at setpoint. For the scope of 
this topic, an "algorithm" is a mathematical relationship between the process variable and setpoint 
inputs of a controller, and the output (manipulated variable). Control algorithms determine how the 
manipulated variable quantity is deduced from PV and SP inputs, and range from the elementary 
to the very complex. In the most common form of control algorithm, the so-called "PID" algorithm, 
calculus is used to determine the proper final control element action for any combination of input 



18.2 On/off control 

Once while working as an instrument technician in a large manufacturing facility, a mechanic asked 
me what it was that I did. I began to explain my job, which was essentially to calibrate, maintain, 
troubleshoot, document, and modify (as needed) all automatic control systems in the facility. The 
mechanic seemed puzzled as I explained the task of "tuning" loop controllers, especially those 
controllers used to maintain the temperature of large, gas-fired industrial furnaces holding many 
tons of molten metal. "Why does a controller have to be 'tuned' ?" he asked. "All a controller does 
is turn the burner on when the metal's too cold, and turn it off when it becomes too hot!" 

In its most basic form, the mechanic's assessment of the control system was correct: to turn 
the burner on when the process variable (molten metal temperature) drops below setpoint, and 
turn it off when it rises above setpoint. However, the actual algorithm is much more complex than 
that, finely adjusting the burner intensity according to the amount of error between PV and SP, 
the amount of time the error has accumulated, and the rate-of-change of the error over time. In 
his limited observation of the furnace controllers, though, he had noticed nothing more than the 
full-on/full-off action of the controller. 

The technical term for a control algorithm that merely checks for the process variable exceeding 
or falling below setpoint is on/off control. In colloquial terms, it is known as bang-bang control, 
since the manipulated variable output of the controller rapidly switches between fully "on" and fully 
"off" with no intermediate state. Control systems this crude usually provide very imprecise control 
of the process variable. Consider our example of the shell-and-tube heat exchanger, if we were to 
implement simple on/off control 1 : 











- — ' 







^-^_ _ 






As you can see, the degree of control is rather poor. The process variable "cycles" between the 
upper and lower setpoints (USP and LSP) without ever stabilizing at the setpoint, because that 

lr To be precise, this form of on/off control is known as differential gap because there are two setpoints with a gap 
in between. While on/off control is possible with a single setpoint (FCE on when below setpoint and off when above), 
it is usually not practical due to the frequent cycling of the final control element. 

18.2. ON/OFF CONTROL 603 

would require the steam valve to be position somewhere between fully closed and fully open. 

This simple control algorithm may be adequate for temperature control in a house, but not for a 
sensitive chemical process! Can you imagine what it would be like if an automobile's cruise control 
system relied on this algorithm? Not only is the lack of precision a problem, but the frequent 
cycling of the final control element may contribute to premature failure due to mechanical wear. 
In the heat exchanger scenario, thermal cycling (hot-cold-hot-cold) will cause metal fatigue in the 
tubes, resulting in a shortened service life. Furthermore, every excursion of the process variable 
above setpoint is wasted energy, because the process fluid is being heated to a greater temperature 
than what is necessary. 

Clearly, the only practical answer to this dilemma is a control algorithm able to proportion the 
final control element rather than just operate it at zero or full effect (the control valve fully closed 
or fully open). This, in its simplest form, is called proportional control. 


18.3 Proportional-only control 

Here is where math starts to enter the algorithm: a proportional controller calculates the difference 
between the process variable signal and the setpoint signal, and calls it the error. This is a measure 
of how far off the process is deviating from its setpoint, and may be calculated as SP-PV or as PV- 
SP, depending on whether or not the controller has to produce an increasing output signal to cause 
an increase in the process variable, or output a decreasing signal to do the same thing. This choice 
in how we subtract determines whether the controller will be reverse-acting or direct- acting. The 
direction of action required of the controller is determined by the nature of the process, transmitter, 
and final control element. In this case, we are assuming that an increasing output signal sent to the 
valve results in increased steam flow, and consequently higher temperature, so our algorithm will 
need to be reverse-acting (i.e. an increase in measured temperature results in a decrease in output 
signal; error calculated as SP-PV). This error signal is then multiplied by a constant value called the 
gain, which is programmed into the controller. The resulting figure, plus a "bias" quantity, becomes 
the output signal sent to the valve to proportion it: 

m = K p e 


m = Controller output 

e = Error (difference between PV and SP) 

K p = Proportional gain 

b = Bias 

If this equation appears to resemble the standard slope-intercept form of linear equation 
(y = mx + 6), it is more than coincidence. Often, the response of a proportional controller is 
shown graphically as a line, the slope of the line representing gain and the y-intercept of the line 
representing the output bias point, or what value the output signal will be when there is zero error 
(PV precisely equals SP): 





1 1 1 1 1 1 1 1 1 1 

+ 10 +20 +30 +40 +50 +60 +70 +80 +90 +100 

Error = (SP - PV) 

In this graph the bias value is 50% and the gain of the controller is 1. 

Proportional controllers give us a choice as to how "sensitive" we want the controller to be to 
changes in process variable (PV) and setpoint (SP). With the simple on/off ("bang-bang") approach, 
there was no adjustment. Here, though, we get to program the controller for any desired level of 

If the controller could be configured for infinite gain, its response would duplicate on/off control. 
That is, any amount of error will result in the output signal becoming "saturated" at either 0% 
or 100%, and the final control element will simply turn on fully when the process variable drops 
below setpoint and turn off fully when the process variable rises above setpoint. Conversely, if the 
controller is set for zero gain, it will become completely unresponsive to changes in either process 
variable or setpoint: the valve will hold its position at the bias point no matter what happens to 
the process. 

Obviously, then, we must set the gain somewhere between infinity and zero in order for this 
algorithm to function any better than on/off control. Just how much gain a controller needs to have 
depends on the process and all the other instruments in the control loop. If the gain is set too high, 
there will be oscillations as the PV converges on a new setpoint value: 






"~R r 


= ^ 

> — 







If the gain is set too low, the process response will be stable under steady-state conditions, but 
"sluggish" to changes in setpoint because the controller does not take aggressive enough action to 
cause quick changes in the process: 








With proportional-only control, the only way to obtain fast-acting response to setpoint changes 
or "upsets" in the process is to set the gain constant high enough that some "overshoot" results: 














As with on/off control, instances of overshoot (the process variable rising above setpoint) and 
undershoot (drifting below setpoint) are generally undesirable, and for the same reasons. Ideally, the 
controller will be able to respond in such a way that the process variable is made equal to setpoint as 
quickly as the process dynamics will allow, yet with no substantial overshoot or undershoot. With 
plain proportional control, however, this ideal goal is nearly impossible. 



18.4 Proportional-only offset 

Another shortcoming of proportional control has to do with changes in process load. A "load" in 
a controlled process is any variable subject to change which has an impact on the variable being 
controlled (the process variable), but is not subject to correction by the controller. In other words, 
a "load" is any variable in the process we cannot or do not control, yet affects the process variable 
we are trying to control. 

In our hypothetical heat exchanger system, the temperature of the incoming process fluid is an 
example of a load: 

Steam in 

Changes in incoming 
feed temperature 
constitute a "load" 
on the process 

Steam out 

If the incoming fluid temperature were to suddenly decrease, the immediate effect this would 
have on the process would be to decrease the outlet temperature (which is the temperature we 
are trying to maintain at a steady value). It should make intuitive sense that a colder incoming 
fluid will require more heat input to raise it to the same outlet temperature as before. If the heat 
input remains the same (at least in the immediate future), this colder incoming flow must make the 
outlet flow colder than it was before. Thus, incoming feed temperature has an impact on the outlet 
temperature whether we like it or not, and the control system has no way to regulate how warm 
or cold the process fluid is before it enters the heat exchanger. This is precisely the definition of a 


Of course, it is the job of the controller to counteract any tendency for the outlet temperature 
to stray from setpoint, but as we shall soon see this cannot be perfectly achieved with proportional 
control alone. 

Let us carefully analyze the scenario of sudden inlet fluid temperature decrease to see how 
a proportional controller would respond. Imagine that previous to this sudden drop in feed 
temperature, the controller was controlling outlet temperature exactly at setpoint (PV = SP) and 
everything was stable. Recall that the equation for a proportional controller is as follows: 

m = K p e + b 


m = Controller output 

e = Error (difference between PV and SP) 

K p = Proportional gain 

b = Bias 

We know that a decrease in feed temperature will result in consequent a decrease of outlet 
temperature with all other factors remaining the same. From the equation we can see that a 
decrease in process variable (PV) will cause the Output value in the proportional controller equation 
to increase. This means a wider-open steam valve, admitting more heating steam into the heat 

All this is good, as we would expect the controller to call for more steam as the outlet temperature 
drops. But will this action be enough to bring the outlet temperature back up to setpoint where it 
was prior to the load change? Unfortunately it will not, although the reason for this may not be 
evident upon first inspection. 

In order to prove that the PV will never go back to SP as long as the incoming feed temperature 
has dropped, let us imagine for a moment that somehow it did. According to the proportional 
controller equation, this would mean that the steam valve would resume its old pre-load-change 
position, only letting through the original flow rate of steam to heat the process fluid. Obviously, if 
the incoming process fluid is colder than before, and the flow rate is unchanged, the same amount 
of heat input (from steam) will result in a colder outlet temperature. In other words, if the steam 
valve goes back to its old position, the outlet temperature will fall just as it did when the incoming 
flow suddenly became colder. This tells us the controller cannot bring the outlet temperature back 
up to setpoint by proportional action alone. 

What will happen is that the controller's output will increase with falling outlet temperature, 
until there is enough steam flow admitted to the heat exchanger to prevent the temperature from 
falling any further. But in order to maintain this greater flow rate of steam (for greater heating 
effect), an error must develop between PV and SP. In other words, the process variable (temperature) 
must fall a bit in order for the controller to call for more steam, in order that the process variable 
does not fall any further than this. 

This necessary error between PV and SP is called proportional- only offset, sometimes less formally 
known as droop. The amount of droop depends on how severe the load change is, and how aggressive 
the controller responds (i.e. how much gain it has). The term "droop" is very misleading, as it 
is possible for the error to develop the other way (i.e. the PV might rise above SP due to a load 
change!). Imagine the opposite load-change scenario, where the incoming feed temperature suddenly 
rises instead of falls. If the controller was controlling exactly at setpoint before this upset, the final 


result will be an outlet temperature that settles at some point above setpoint, enough so that the 
controller is able to pinch the steam valve far enough closed to stop any further rise in temperature. 

We can minimize proportional-only offset by increasing controller gain. This makes the controller 
more "aggressive" so that it moves the control valve further for any given change in PV or SP. Thus, 
not as much error needs to develop between PV and SP to move the valve to any new position it 
needs to go. However, too much gain and the control system will begin to oscillate just like a crude 
on/off controller. 

If we are limited in how much gain we can program in to the controller, how do we minimize 
this offset? One way is for a human operator to periodically place the controller in manual mode 
and move the control valve just a little bit more so that the PV once again reaches SP, then place 
the controller back into automatic mode. In essence this technique adjusts the "Bias" term of 
the controller equation. The disadvantage of this technique is rather obvious: it requires frequent 
human intervention. What's the point of having an automation system that needs periodic human 
intervention to maintain setpoint? 

A more sophisticated method for eliminating proportional-only offset is to add a different control 
action to the controller: one that takes action based on the amount of error between PV and SP 
and the amount of time that error has existed. We call this control mode integral, or reset. This 
will be the subject of the next section. 


18.5 Integral (reset) control 

Integration is a calculus principle, but don't let the word "calculus" scare you. You are probably 
already familiar with the concept of numerical integration even though you may have never heard 
of the term before. 

Calculus is a form of mathematics that deals with changing variables, and how rates of change 
relate between different variables. When we "integrate" a variable with respect to time, what we 
are doing is accumulating that variable's value as time progresses. Perhaps the simplest example 
of this is a vehicle odometer, which accumulates the total distance traveled by the vehicle over a 
certain time period. This stands in contrast to a speedometer, which indicates how far the vehicle 
is traveling per unit of time. 

Imagine a car moving along at exactly 30 miles per hour. How far will this vehicle travel after 1 
hour of driving this speed? Obviously, it will travel 30 miles. Now, how far will this vehicle travel 
if it continues for another 2 hours at the exact same speed? Obviously, it will travel 60 more miles, 
for a total distance of 90 miles since it began moving. If the car's speed is a constant, calculating 
total distance traveled is a simple matter of multiplying that speed by the travel time. 

The odometer mechanism that keeps track of the mileage traveled by the car may be thought of 
as integrating the speed of the car with respect to time. In essence, it is multiplying speed times 
time continuously to keep a running total of how far the car has gone. When the car is traveling 
at a high speed, the odometer "integrates" at a faster rate. When the car is traveling slowly, the 
odometer "integrates" slowly. 

If the car travels in reverse, the odometer will decrement (count down) rather than increment 
(count up) because it sees a negative quantity for speed 2 . The rate at which the odometer decrements 
depends on how fast the car travels in reverse. When the car is stopped (zero speed), the odometer 
holds its reading and neither increments nor decrements. 

Now imagine how this concept might apply to a process controller. Integration is provided 
either by a mechanism (in the case of a pneumatic controller) , an op-amp circuit (in the case of an 
analog electronic controller), or by a microprocessor running a digital integration algorithm. The 
variable being integrated is error (the difference between PV and SP). Thus the integral mode of 
the controller ramps the output either up or down over time, the direction of ramping determined 
by the sign of the error (PV greater or less than SP), and the rate of ramping determined by the 
magnitude of the error (how far away PV is from SP). 

If proportional action is where the error tells the output how far to move, integral action is where 
the error tells the output how fast to move. One might think of integral as being how "impatient" 
the controller is, with integral action constantly ramping the output as far as it needs to go in order 
to eliminate error. Once the error is zero (PV = SP), of course, the integral action stops ramping, 
leaving the controller output (valve position) at its last value just like a stopped car's odometer 
holds a constant value. 

If we add an integral term to the controller equation, we get something that looks like this: 

in = K p e + Ki / e dt + b 

2 At least the old-fashioned mechanical odometers would. Some new cars use a pulse detector on the driveshaft 
which cannot tell the difference between forward and reverse, and therefore their odometers always increment. Shades 
of Ferris Bueller's Day Off. 



m = Controller output 

e = Error (difference between PV and SP) 

K p = Proportional gain 

Ki = Integral gain 

t = Time 

b = Bias 

The most confusing portion of this equation for those new to calculus is the part that says 
"/ e dt" . The integration symbol (looks like an elongated letter "S" ) tells us the controller will 
accumulate ("sum") multiple products of error (e) over tiny slices of time (dt). Quite literally, the 
controller multiplies error by time (for very short segments of time) and continuously adds up all 
those products to contribute to the output signal which then drives the control valve (or other final 
control element). 

To see how this works in a practical sense, let's imagine how a proportional + integral controller 
would respond to the scenario of a heat exchanger whose inlet temperature suddenly dropped. As 
we saw with proportional-only control, an inevitable offset occurs between PV and SP with changes 
in load, because an error must develop if the controller is to generate the different output signal 
value necessary to halt further change in PV. 

Once this error develops, though, integral action begins to work. Over time, a larger and larger 
quantity accumulates in the integral mechanism (or register) of the controller because an error 
persists over time. That accumulated value adds to the controller's output, driving the steam control 
valve further and further open. This, of course, adds heat at a faster rate to the heat exchanger, 
which causes the outlet temperature to rise. As the temperature re-approaches setpoint, the error 
becomes smaller and thus the integral action proceeds at a slower rate (like a car's odometer ticking 
by at a slower rate when the car's speed decreases). So long as the PV is below SP (the outlet 
temperature is still too cool), the controller will continue to integrate upwards, driving the control 
valve further and further open. Only when the PV rises to exactly meet SP does integral action 
finally rest, holding the valve at a steady position. 

Integral is a highly effective mode of process control. In fact, some processes respond so well 
to integral controller action that it is possible to operate the control loop on integral action alone, 
without proportional. Typically, though, process controllers are designed to operate as proportional- 
only (P), proportional plus integral (PI). 

Just as too much proportional gain will cause a process control system to oscillate, too much 
integral gain will also cause oscillation. If the integration happens at too fast a rate, the controller's 
output will "saturate" either high or low before the process variable can make it back to setpoint. 
Once this happens, the only condition that will "unwind" the accumulated integral quantity is for 
an error to develop of the opposite sign, and remain that way long enough for a canceling quantity 
to accumulate. Thus, the PV must cross over the SP, guaranteeing at least another half-cycle of 

A similar problem called reset windup (or integral windup) happens when external conditions 
make it impossible for the controller to hold the process variable equal to setpoint. Imagine what 
would happen in the heat exchanger system if the steam boiler suddenly stopped producing steam. 
As outlet temperature dropped, the controller's proportional action would open up the control valve 


in a futile effort to raise temperature. If and when steam service is restored, proportional action 
would just move the valve back to its original position as the process variable returned to its original 
value (before the boiler died). This is how a proportional-only controller would respond to a steam 
"outage": nice and predictably. If the controller had integral action, however, a much worse condition 
would result. All the time spent with the outlet temperature below setpoint causes the controller's 
integral term to "wind up" in a futile attempt to admit more steam to the heat exchanger. This 
accumulated quantity can only be un-done by the process variable rising above setpoint for an equal 
error-time product, which means when the steam supply resumes, the temperature will rise well 
above setpoint until the integral action finally "unwinds" and brings the control valve back to a sane 
position again. 

Various techniques exist to manage integral windup. Controllers may be built with limits to 
restrict how far the integral term can accumulate under adverse conditions. In some controllers, 
integral action may be turned off completely if the error exceeds a certain value. The surest fix for 
integral windup is human operator intervention, by placing the controller in manual mode. This 
typically resets the integral accumulator to a value of zero and loads a new value into the bias term 
of the equation to set the valve position wherever the operator decides. Operators usually wait until 
the process variable has returned at or near setpoint before releasing the controller into automatic 
mode again. 

While it might appear that operator intervention is again a problem to be avoided (as it was 
in the case of having to correct for proportional-only offset), it is noteworthy to consider that 
the conditions leading to integral windup usually occur only during shut-down conditions. It is 
customary for human operators to run the process manually anyway during a shutdown, and so the 
switch to manual mode is something they would do anyway and the potential problem of windup 
often never manifests itself. 


18.6 Derivative (rate) control 

The final facet of PID control is the "D" term, which stands for derivative. This is a calculus 
concept like integral, except most people consider it easier to understand. Simply put, derivative is 
the expression of a variable's rate- of- change with respect to another variable. Finding the derivative 
of a function (differentiation) is the inverse operation of integration. With integration, we calculated 
accumulated value of some variable's product with time. With derivative, we calculate the ratio of a 
variable's change per unit of time. Whereas integration is fundamentally a multiplicative operation 
(products), differentiation always involves division (ratios). 

A controller with derivative (or rate) action looks at how fast the process variable changes per 
unit of time, and takes action proportional to that rate of change. In contrast to integral (reset) 
action which represents the "impatience" of the controller, derivative (rate) action represents the 
"cautious" side of the controller. 

If the process variable starts to change at a high rate of speed, the job of derivative action is to 
move the control valve in such a direction as to counteract this rapid change, and thereby moderate 
the speed at which the process variable changes. 

What this will do is make the controller "cautious" with regard to rapid changes in process 
variable. If the process variable is headed toward the setpoint value at a rapid rate, the derivative 
term of the equation will diminish the output signal, thus slowing tempering the control response 
and slowing the process variable's approach toward setpoint. To use an automotive analogy, it is 
as if a driver, driving a very heavy vehicle, preemptively applies the brakes to slow the vehicle's 
approach to an intersection, knowing that the vehicle doesn't "stop on a dime." The heavier the 
vehicle, the sooner a wise driver will apply the brakes, to avoid "overshoot" beyond the stop sign 
and into the intersection. 

If we modify the controller equation to incorporate differentiation, it will look something like 

f de 

m = K p e + Ki I edt + K d — + b 


m = Controller output 

e = Error (difference between PV and SP) 

K p = Proportional gain 

Ki = Integral gain 

Kd = Derivative gain 

t = Time 

b = Bias 

The 4| term of the equation expresses the rate of change of error (e) over time (t). The lower-case 
letter "d" symbols represent the calculus concept of differentials which may be thought of in this 
context as very tiny increments of the following variables. In other words, 4| refers to the ratio 
of a very small change in error (de) over a very small increment of time (di). On a graph, this is 
interpreted as the slope of a curve at a specific point (slope being defined as rise over run). 

It should be mentioned that derivative mode should be used with caution. Since it acts on rates 
of change, derivative action will "go crazy" if it sees substantial noise in the PV signal. Even small 
amounts of noise possess extremely large rates of change (defined as percent PV change per minute 


of time) owing to the relatively high frequency of noise compared to the timescale of physical process 


18.7 PID controller tuning 

So far we have seen three different controller actions which may be applied to stabilize an automated 
process: proportional, integral, and derivative, or PID. The relative effect of each action in a controller 
may be set by the instrument technician, by adjusting the values of K p , Ki, and Kd- The act of 
adjusting these three gain values to achieve optimum control stability is called tuning. 

In a mechanical (pneumatic) PID controller, these constants are typically adjusted by manually 
moving fulcrum positions and needle valve positions. In analog electronic PID controllers, 
potentiometers and switch-selectable capacitors typically control the gain settings. Digital electronic 
controllers, of course, are simply programmed with direct numerical values for K p , Ki, and Kd- 

A very unfortunate source of confusion in the world of PID controllers is different units for 
expressing P, I, and D constants. Beginning with proportional, we have two ways of expressing 
the "aggressiveness" of the controller: gain and proportional band. Gain is exactly what you might 
expect it to be if you have an electronics background: a direct ratio of output change to input 
change. For a proportional-only controller, a gain of 2 means that a 10% change in error results in a 
20% change in output signal. The other way of expressing proportional action, called "proportional 
band" defines the controller's aggressiveness in terms of how much input change is necessary (in 
units of percent) to produce a full-scale (100%) change in output signal. Thus, a gain of 2 would be 
expressed as a proportional band of 50%. A gain of 5 would be equivalent to a proportional band 
of 20%. A gain of 0.4 is the same as a proportional band of 250%. These are just two different 
(reciprocal) ways of saying the same thing. 

Integral isn't any better. We have two (reciprocal, again) ways of expressing how fast a controller 
will ramp its output (integrate) given a constant input error. The first way is in units of time, usually 
minutes or seconds. An integral-only controller with a tuning constant value of 2 minutes and a 
constant error (difference between SP and PV) of 10% will ramp the output at a constant rate of 10% 
every 2 minutes, or 5% per minute. Often, we find integral action included with proportional in the 
same controller, and so the integral constant is sometimes expressed in minutes per repeat instead of 
just minutes, referring to how many minutes the integral action will "repeat" proportional's action. 
Thus, a PI controller with a proportional gain of 1, an integral constant of 5 minutes per repeat, 
and a constant error of 10% will take 5 minutes to ramp the output 10% (or 2% per minute). The 
same controller with twice the proportional gain will only take 2.5 minutes to ramp the output the 
same amount (4% per minute) if the algorithm is such that proportional gain influences both P and 
I terms (which is quite common). One could also express that controller's integral action as 0.2 
repeats per minute instead of 5 minutes per repeat, just to be confusing. 

Derivative is the most consistent tuning parameter of them all, always being expressed directly 
in units of time, usually minutes or seconds. A controller with a derivative time setting of 2 minutes 
will generate an output offset of 10% if it sees the error changing at a steady rate of 5% per minute. 
That is, unless the proportional gain of the controller also affects derivative action, in which case 
the amount of offset introduced by derivative action will be multiplied by the gain value. 

Oh, but the fun doesn't end here. In addition to having multiple units of measurement to express 
PID settings, we also have multiple algorithms for calculating the controller output. The version 
I've been showing you thus far in this section is called the parallel algorithm, with the P, I, and D 
terms all separate: 


K p e + K % edt + K d 


If only things were always this simple! As luck would have it, the best algorithm for tuning real 
processes (called the ISA algorithm) uses K p as a multiplier for all three terms, and so the resulting 
equation looks different: 

Kpie + Ki e dt + K d -^j + b 

Back in the days when pneumatic controllers were the norm, it was expensive to build PID 
controllers to implement either of these two equations, and so an equation form better suited for 
mechanical design became popular. Known as the "series" PID algorithm, its equation looks like 

m = K p (e + Ki edt) ( 1 + K d — 

Even though pneumatic and analog electronic PID controllers are mostly obsolete, we still see 
the old "series" algorithm implemented in some modern digital controllers for the sake of direct 
interchangeability. This way, someone can upgrade their old control system and use the exact same 
P, I, and D tuning constants in the new controller to control the process just as well. 

Just in case you thought things still weren't complicated enough, we have even more variations 
on PID control algorithms to consider. Some controllers calculate the derivative term on error, while 
others calculate it on process variable alone. The difference in response between these two controller 
types is revealed when a human operator makes a setpoint change: the PV-based derivative controller 
does nothing, while the error-based derivative controller makes a sudden "jump" in its output value. 

Furthermore, many digital electronic controllers calculate the PID equation based on changes in 
PV rather than the absolute value of PV. This is known as the velocity algorithm, as opposed to 
the position algorithm. The difference between these two variations of PID control becomes evident 
when one changes the gain setting: the velocity algorithm controller does nothing, while the position 
algorithm controller makes a sudden "jump" in its output value. 

Most digital electronic controllers also have provision for process variable filtering, the purpose of 
which is to dampen unwanted noise from the PV signal. Over-enthusiastic use of filtering, however, 
can cause major problems with PID control. Too much filtering, and the PID algorithm does not 
see the "real-time" value of the PV, and consequently will begin to control a sluggish version of the 
PV instead of the real PV. 

There is much that may be said about controller tuning. The first and most significant point 
to be made about tuning is this: don't, unless you know what you are doing. Poor control caused 
by improper tuning is very wasteful of product and energy in a manufacturing operation, and can 
even be dangerous to operations if too unstable. Far too many control loops run erratically because 
someone decided to mess with the controller's PID settings when they did not understand how or 
why to do so. 

When a formerly robust control system gets "out of tune," the cause is almost always due 
to problems in the process or field instrumentation. Unfortunately, what a lot of good-intended 
technicians do is go straight to the controller and try adjusting the P, I, and D settings because it 



looks easy and it will make them look really smart to be able to fix the problem just by making a 
few technical adjustments. No amount of P, I, or D setting adjustments, though, will correct for 
actual process or equipment failures. The real solution is to diagnose the process to determine the 
root cause of the instability. 

Ironically, one of the best diagnostic tools available to the technician is the controller's manual 
mode. By placing the controller in manual mode, we "disconnect" the input from the output so the 
output no longer responds to changes in PV or SP. This breaks the feedback loop of the system so 
that it has a definite beginning and end: 

Input Output 


(manual mode) 



The Process 

By making carefully measured changes in the controller's output and examining the consequent 
changes in transmitter signal, one may determine the existence of many different problems including 
transmitter and process noise, improper transmitter ranges, final control element hysteresis, 
variations in process gain, lag times, dead times, and other impediments to optimum control. Only 
after examining the process and its response to changes in valve position should controller tuning 
be attempted. 



Lavigne, John R., Instrumentation Applications for the Pulp and Paper Industry, The Foxboro 
Company, Foxboro, MA, 1979. 

Liptak, Bela G., Instrument Engineers' Handbook - Process Control Volume II, Third Edition, CRC 
Press, Boca Raton, FL, 1999. 

Shinskey, Francis G., Energy Conservation through Control, Academic Press, New York, NY, 1978. 

Shinskey, Francis G., Process- Control Systems - Application / Design / Adjustment, Second Edition, 
McGraw-Hill Book Company, New York, NY, 1979. 

St. Clair, David W., Controller Tuning and Control Loop Performance, a primer, Straight-Line 
Control Company, Newark, DE, 1989. 


Appendix A 

Doctor Strangeflow, or how I 
learned to relax and love Reynolds 


Of all the non-analytical (non-chemistry) process measurements students encounter in their 
Instrumentation training, flow measurement is one of the most mysterious. Where else would we have 
to take the square root of a transmitter signal just to measure a process variable in the simplest case? 
Since flow measurement is so vital to many industries, it cannot go untouched in an Instrumentation 
curriculum. Students must learn how to measure flow, and how to do it accurately. The fact that it 
is a fundamentally complex thing, however, often leads to oversimplification in the classroom. Such 
was definitely the case in my own education, and it lead to a number of misunderstandings that 
were corrected after a lapse of 15 years, in a sudden "Aha!" moment that I now wish to share with 

The orifice plate is to flow measurement what a thermocouple is to temperature measurement: 
an inexpensive yet effective primary sensing element. The concept is disarmingly simple. Place a 
restriction in a pipe, then measure the resulting pressure drop (AP) across that restriction to infer 
flow rate. You may have already seen a diagram such as the following, illustrating how an orifice 
plate works: 







Direction of flow 

Now, the really weird thing about measuring flow this way is that the resulting AP signal does 
not linearly correspond to flow rate. Double the flow rate, and the AP quadruples. Triple the flow 
rate and the AP increases by a factor of nine. To express this relationship mathematically: 

) 2 <x AP 

In other words, differential pressure across an orifice plate (AP) is proportional to the square of 
the flow rate (Q 2 ). To be more precise, we may include a coefficient (k) with a precise value that 
turns the proportionality into an equality: 

Q 2 = fc(AP) 

Expressed in graphical form, the function looks like one-half of a parabola: 


Diff. pressure 

Flow (Q) 

To obtain a linear flow measurement signal from the differential pressure instrument's output 
signal, we must "square root" that signal, either with a computer inside the transmitter, with 
a computer inside the receiving instrument, or a separate computing instrument (a "square root 
extractor"). We may see mathematically how this yields a value for flow rate (Q), following from 
our original equation: 

Q 2 = k{AP) 


Q = ^k[AP) 

substituting a new coefficient value k 1 

Students are taught that the differential pressure develops as a consequence of energy conservation 
in the flowing liquid stream. As the liquid enters a constriction, its velocity must increase to account 
for the same volumetric rate through a reduced area. This results in kinetic energy increasing, which 
must be accompanied by a corresponding decrease in potential energy (i.e. pressure) to conserve 
total fluid energy. Pressure measurements taken in a venturi pipe confirm this: 

1 Since we get to choose whatever k value we need to make this an equality, we don't have to keep k inside the 
radicand, and so you will usually see the equation written as it is shown in the last step with k outside the radicand. 


High pressure 

High pressure 

Low velocity 

Low velocity 

In all honesty, this did not make sense to me when I heard this. My "common sense" told me 
the fluid pressure would increase as it became crammed into the constriction, not decrease. Even 
more, "common sense" told me that whatever pressure was lost through the constriction would never 
be regained, contrary to the pressure indication of the gauge furthest downstream. Accepting this 
principle was an act of faith on my part, putting preconceived notions aside for something new. A 
leap of faith, however, is not the same as a leap in understanding. I believed what I was told, but I 
really didn't understand why it was true. 

The problem intensified when my teacher showed a more detailed flow equation. This new 
equation contained a term for fluid density (p): 

Q = k\ 


What this equation showed us is that orifice plate flow measurement depended on density. If 
the fluid density changed, our instrument calibration would have to change in order to maintain 
good accuracy of measurement. Something disturbed me about this equation, though, so I raised 
my hand. The subsequent exchange between me and my teacher went something like this: 

Me: What about viscosity? 

Teacher: What? 

Me: Doesn't fluid viscosity have an effect on flow measurement, just like density? 

Teacher: You don't see a variable for viscosity in the equation, do you? 

Me: Well, no, but it's got to have some effect on flow measurement! 

Teacher: How come? 

Me: Imagine clean water flowing through a venturi, or through the hole of an orifice plate. 
At a certain flow rate, a certain amount of AP will develop across the orifice. Now imagine 


that same orifice flowing an equal rate of liquid honey: approximately the same density as 
water, but much thicker. Wouldn't the increased "thickness," or viscosity, of the honey result 
in more friction through the orifice, and thus more of a pressure drop than what the water 
would create? 

Teacher: I'm sure viscosity has some effect, but it must be minimal since it isn't in the 

Me: Then why is honey so hard to suck through a straw? 

Teacher: Come again? 

Me: A straw is a narrow pipe, similar to the throat of a venturi or the hole of an orifice, 
right? The difference in pressure between the suction in my mouth and the atmosphere is 
the AP across that orifice. The result is flow through the straw. If viscosity is of such little 
effect, then why is liquid honey so much harder to suck through a straw than water? The 
pressure is the same, the density is about the same, then why isn't the flow rate the same 
according to the equation you just gave us? 

Teacher: In industry, we usually don't measure fluids as thick as honey, and so it's safe to 
ignore viscosity in the flow equation . . . 

My teacher's smokescreen - that thick fluid flow streams were rare in industry - did nothing to 
alleviate my confusion. Despite my ignorance of the industrial world, I could very easily imagine 
liquids that were more viscous than water, honey or no honey. Somewhere, somehow, someone had 
to be measuring the flow rate of such liquids, and there the effects of viscosity on orifice AP must 
be apparent. Surely my teacher knew this. But then why did the flow equation not have a variable 
for viscosity in it? How could this parameter be unimportant? Like most students, though, I could 
see that arguing would get me nowhere and it was better for my grade to just go along with what 
the teacher said than to press for answers he couldn't give. In other words, I swept my doubts under 
the carpet of "learning" and made a leap of faith. 

After that, we studied different types of orifice plates, different types of pressure tap locations, 
and other inferential primary sensing elements (annubars, target meters, pipe elbows, etc.). They 
all worked on Bernoulli's principle of decreased pressure through a restriction, and they all required 
square root extraction of the pressure signal to obtain a linearized flow measurement. In fact, this 
became the sole criterion for determining whether or not we needed square root extraction on the 
signal: did the flow measurement originate from a differential pressure instrument? If so, then we 
needed to "square root" the signal. If not, we didn't. A neat and clean distinction, separating AP- 
based flow measurements from all the others (magnetic, vortex shedding, Coriolis effect, thermal, 
etc.). Nice, clean, simple, neat, and only 95% correct, as I was to discover later. 

Fast-forward fifteen years. I was now a teacher in a technical college, teaching Instrumentation 
to students just like myself a decade and a half ago. It was my first time preparing to teach flow 
measurement, and so I brushed up on my knowledge by consulting one of the best technical references 
I could get my hands on: Bela Liptak's Process Measurement and Analysis, third edition. Part of 
the Instrument Engineers' Handbook series, this wonderful work was to be our primary text as we 


explored the world of process measurement during the 2002-2003 academic year. 

It was in reading this book that I had an epiphany. Section 2.8 of the text discussed a type of 
flowmeter I had never seen or heard of before: the laminar flowmeter. As I read this section of 
the book, my jaw hit the floor. Here was a differential-pressure-based flowmeter that was linear! 
That is, there was no square root extraction required at all to convert the AP measurement into a 
flow measurement. Furthermore, its operation was based on some weird equation called the Hagen- 
Poiseuille Law rather than Bernoulli's Law. 

Early in the section's discussion of this flowmeter, a couple of paragraphs explained the meaning 
of something called Reynolds number of a flow stream, and how this was critically important to 
laminar flowmeters. Now, I had heard of Reynolds number before when I worked in industry, but I 
never knew what it meant. All I knew is that it had something to do with the selection of flowmeter 
types: one must know the Reynolds number of a fluid before one could properly select which type 
of flow-measuring instrument to use in a particular application. Since this determination typically 
fell within the domain of instrument engineers and not instrument technicians (as I was), I gave 
myself permission to remain ignorant about it and blissfully went on my way. Little did I know that 
Reynolds number held the key to understanding my "honey-through-a-straw" question of years ago, 
as well as comprehending (not just believing) how orifice plates actually worked. 

According to Liptak, laminar flowmeters were effective only for low Reynolds numbers, typically 
below 1200. Cross-referencing the orifice plate section of the same book told me that Reynolds 
numbers for typical orifice-plate flow streams were much greater (10,000 or higher). Furthermore, 
the orifice plate section contained an insightful passage on page 152 which I will now quote here. 
Italicized words indicate my own emphasis, locating the exact points of my "Aha!" moments: 

The basic equations of flow assume that the velocity of flow is uniform across a given 
cross-section. In practice, flow velocity at any cross section approaches zero in the boundary 
layer adjacent to the pipe wall, and varies across the diameter. This flow velocity profile has a 
significant effect on the relationship between flow velocity and pressure difference developed in 
a head meter. In 1883, Sir Osborne Reynolds, an English scientist, presented a paper before 
the Royal Society, proposing a single, dimensionless ratio now known as Reynolds number, 
as a criterion to describe this phenomenon. This number, Re, is expressed as 

Re= V -^ 

where V is velocity, D is diameter, p is density, and /z is absolute viscosity. Reynolds number 
expresses the ratio of inertial forces to viscous forces. At a very low Reynolds number, viscous 
forces predominate, and the inertial forces have little effect. Pressure difference approaches 
direct proportionality to average flow velocity and to viscosity. At high Reynolds numbers, 
inertial forces predominate and viscous drag effects become negligible. 

What the second paragraph is saying is that for slow-moving, viscous fluids (such as honey in a 
straw) , the forces of friction (fluid "dragging" against the pipe walls) are far greater than the forces 
of inertia (fluid momentum). This means that the pressure difference required to move such a fluid 
through a pipe primarily works to overcome the friction of that fluid against the walls of the pipe. 
For most industrial flows, where the flow velocities are fast and the fluids have little viscosity (like 
clean water), flow through an orifice plate is assumed to be frictionless. Thus, the pressure dropped 
across a constriction is not the result of friction between the fluid and the pipe, but rather it is 
a consequence of having to accelerate the fluid from a low velocity to a high velocity through the 
narrow orifice. 


My mistake, years ago, was in assuming that water flowing through an orifice generated 
substantial friction, and that this is what created the AP across an orifice plate. This is what 
my "common sense" told me. In my mind, I imagined the water having to rub past the walls of 
the pipe, past the face of the orifice plate, and through the constriction of the orifice at a very high 
speed, in order to make it through to the other side. I memorized what my teacher told us about 
energy exchange and how pressure had to drop as velocity increased, but I never really internalized 
it because I still held to my faulty assumption of friction being the dominant mechanism of pressure 
drop in an orifice plate. In other words, while I could parrot the doctrine of kinetic and potential 
energy exchange, I was still thinking in terms of friction, which is a totally different phenomenon. 
The difference between these two phenomena is the difference between energy exchanged and energy 
dissipated. To use an electrical analogy, it is the difference between reactance (X) and resistance (R). 
Incidentally, many electronics students experience the same confusion when they study reactance, 
mistakenly thinking it is the same thing as resistance where in reality it is quite different in terms 
of energy, but that is a subject for another essay! 

In a frictionless flow stream, fluid pressure decreases as fluid velocity increases in order to conserve 
energy. Another way to think of this is that a pressure differential must develop in order to provide 
the "push" needed to accelerate the fluid from a low speed to a high speed. Conversely, as the fluid 
slows back down after having passed through the constriction, a reverse pressure differential must 
develop in order to provide the "push" needed for that deceleration: 

Direction of "push" 

Direction of "push" 

Low velocity 

Low velocity 

A moving mass does not simply slow down on its own! There must be some opposing force to 
decelerate a mass from a high speed to a low speed. This is where the pressure recovery downstream 
of the orifice plate comes from. If the pressure differential across an orifice plate originated primarily 
from friction, as I mistakenly assumed when I first learned about orifice plates, then there would be 
no reason for the pressure to ever recover downstream of the constriction. The presence of friction 
means energy lost, not energy exchanged. Although both inertia and friction are capable of creating 
pressure drops, the lasting effects of these two different phenomena are definitely not the same. 

There is a quadratic ("square") relationship between velocity and differential pressure precisely 
because there is a quadratic relationship between velocity and kinetic energy as all first-quarter 

physics students learn (E^ 

This is why AP increases with the square of flow rate (Q ) 


and why we must "square-root" the AP signal to obtain a flow measurement. This is also why fluid 
density is so important in the orifice-plate flow equation. The denser a fluid is, the more work will be 
required to accelerate it through a constriction, resulting in greater AP, all other conditions being 


Q = k\l (Our old friend, the "orifice plate" equation) 


This equation is only accurate, however, when fluid friction is negligible: when the viscosity 
of the fluid is so low and/or its speed is so high that the effects of potential and kinetic energy 
exchange completely overshadow 2 the effects of friction against the pipe walls and against the orifice 
plate. This is indeed the case for most industrial flow applications, and so this is what students first 
study as they learn how flow is measured. Unfortunately, this is often the only equation two-year 
Instrumentation students study with regard to flow measurement. 

In situations where Reynolds number is low, fluid friction becomes the dominant factor and 
the standard "orifice plate" equation no longer applies. Here, the AP generated by a viscous 
fluid moving through a pipe really does depend primarily on how "thick" the fluid is. And, just like 
electrons moving through a resistor in an electric circuit, the pressure drop across the area of friction 
is directly proportional to the rate of flow (AP oc Q for fluids, V oc I for electrons). This is why 
laminar flowmeters - which work only when Reynolds number is low - yield a nice linear relationship 
between AP and flow rate and therefore do not require square root extraction of the AP signal. 
These flowmeters do, however, require temperature compensation (and even temperature control 
in some cases) because flow measurement accuracy depends on fluid viscosity, and fluid viscosity 
varies according to temperature. The Hagen-Poiseuille equation describing flow rate and differential 
pressure for laminar flow (low Re) is shown here for comparison: 


Q = Flow rate (gallons per minute) 

k = Unit conversion factor = 7.86 x 10 5 

AP = Pressure drop (inches of water column) 

D = Pipe diameter (inches) 

fi = Liquid viscosity (centipoise) - this is a temperature-dependent variable! 

L = Length of pipe section (inches) 

Note that if the pipe dimensions and fluid viscosity are held constant, the relationship between 
flow and differential pressure is a direct proportion: 

Qoc AP 

2 In engineering, this goes by the romantic name of swamping. We say that the overshadowing effect "swamps" out 
all others because of its vastly superior magnitude, and so it is safe (not to mention simpler!) to ignore the smaller 
effect (s). The most elegant cases of "swamping" are when an engineer intentionally designs a system so that the 
desired effect is many times greater than the undesired effect (s), thereby forcing the system to behave more like the 
ideal. This application of swamping is prevalent in electrical engineering, where resistors are often added to circuits 
for the purpose of overshadowing the effects of stray (undesirable) resistance in wiring and components. 


In reality, there is no such thing as a frictionless flow (excepting superfluidic cases such as 
Helium II which are well outside the bounds of normal experience), just as there is no such thing as 
a massless flow (no inertia). In normal applications there will always be both effects at work. By 
not considering fluid friction for high Reynolds numbers and not considering fluid density for low 
Reynolds numbers, engineers draw simplified models of reality which allow us to more easily measure 
fluid flow. As in so many other areas of study, we exchange accuracy for simplicity, precision for 
convenience. Problems arise when we forget that we've made this Faustian exchange and wander 
into areas where our simplistic models are no longer accurate. 

Perhaps the most practical upshot of all this for students of Instrumentation is to realize exactly 
why and how orifice plates work. Bernoulli's equation does not include any considerations of friction. 
To the contrary, we must assume the fluid to be completely frictionless in order for the concept to 
make sense. This explains several things: 

• There is little permanent pressure drop across an orifice: most of the pressure lost at the vena 
contracta is regained further on downstream as the fluid returns to its original (slow) speed. 
Permanent pressure drop will occur only where there is energy lost through the constriction, 
such as in cases where fluid friction is substantial. Where the fluid is frictionless there is no 
mechanism in an orifice to dissipate energy, and so with no energy lost there must be full 
pressure recovery as the fluid returns to its original speed. 

• Pressure tap location makes a difference: to ensure that the downstream tap is actually sensing 
the pressure at a point where the fluid is moving significantly faster than upstream (the "vena 
contracta"), and not just anywhere downstream of the orifice. If the pressure drop were due 
to friction alone, it would be permanent and the downstream tap location would not be as 

• Standard orifice plates have knife-edges on their upstream sides: to minimize contact area 
(friction points) with the high-speed flow. 

• Care must be taken to ensure Reynolds number is high enough to permit the use of an orifice 
plate: if not, the linear Q/AP relationship for viscous flow will assert itself along with the 
quadratic potential/kinetic energy relationship, causing the overall Q/AP relationship to be 
polynomial rather than purely quadratic, and thereby corrupting the measurement accuracy. 

• Sufficient upstream pipe length is needed to condition flow for orifice plate measurement, not 
to make it "laminar" as is popularly (and wrongly) believed, but to allow natural turbulence 
to "flatten" the flow profile for uniform velocity. Laminar flow is something that only happens 
when viscous forces overshadow inertial forces (e.g. flow at low Reynolds numbers), and is 
totally different from the fully developed turbulent flow that orifice plates need for accurate 

In a more general sense, the lesson we should learn here is that blind faith is no substitute for 
understanding, and that a sense of confusion or disagreement during the learning process is a sign 
of one or more misconceptions in need of correction. If you find yourself disagreeing with what you 
are being taught, either you are making a mistake and/or your teacher is. Pursuing your questions 
to their logical end is the key to discovery, while making a leap of faith (simply believing what you 
are told) is an act of avoidance: escaping the discomfort of confusion and uncertainty at the expense 
of a deeper learning experience. This is an exchange no student should feel they have to make. 



Liptak, Bela G., Instrument Engineers' Handbook - Process Measurement and Analysis Volume I, 
Third Edition, CRC Press, New York, NY. 

Appendix B 

Creative Commons Attribution 



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j operator, 121 
10 to 50 mA, 201 

3 to 15 PSI, 210 
3-valve manifold, 322 

4 to 20 mA, 187, 209 

4-wire resistance measurement circuit, 424, 

4-wire transmitter, 198 
5-point calibration, 265 
5-valve manifold, 323 

Absolute pressure, 43, 294 

Absolute viscosity, 49 

AC excitation, magnetic flowmeter, 510 

Acid, 70, 71 

Activation energy, 65 

Address, 252 

AGA Report #3, 479, 484 

AGA Report #7, 499 

AGA Report #9, 513 

Algorithm, 601 

Algorithm, control, 156 

Alkaline, 70, 71 

American Gas Association, 479, 484, 499, 513 

Ampere, Andre, 80 

Ampere, 80 

Analyzer, 285 

Angle of repose, 392 

Anion, 542 

Annubar, 467, 532 

Anode, 542 

API, degrees, 38 

Archimedes' Principle, 45, 385 

As-found calibration, 270 

As-left calibration, 270 

Atmospheres, 44 

Atom, 61 

Atomic clock, 271 
Atomic mass, 62 
Atomic number, 62 
Atomic weight, 62 
Automatic mode, 131 
Averaging Pitot tube, 467 
Avogadro's number, 63 

B.I.F. Universal Venturi tube, 469 

Backpressure, nozzle, 213 

Baffle, 213 

Balance beam scale, 216 

Balling, degrees, 39 

Bang-bang control, 602 

Bark, degrees, 39 

Barometer, 294 

Base, 70, 71 

Base unit, 19 

Baume, degrees, 38 

Bellows, 217, 295 

Bernoulli's equation, 55, 450 

Bernoulli, Daniel, 55 

Beta ratio of flow element, 483 

Bethlehem flow tube, 469 

Bi-metal strip, 417 

Biological oxygen demand, 601 

Blackbody, 435 

Blackbody calibrator, 277 

Bleed valve fitting, 326 

Bluff body, 501 

BOD, 601 

Boiling point of water, 275 

Bourdon tube, 295 

Boyle's Law, 47 

Bridge circuit, 100 

Brix, degrees, 39 

Bubble tube, 361 




Buffer solution, 285, 550, 560 
Buoyancy, 45, 385 
Buoyant test of density, 46 
Burnout, thermocouple, 433 

Calibration, 257, 285 

Calibration gas, 287 

Capacitance, 108 

Capacitor, 108 

Capillary tube, 333, 486 

Cathode, 542 

Cation, 542 

Caustic, 70, 71 

Celsius, 275 

Centigrade, 275 

Centrifugal force, 515 

Centripetal force, 515 

cgs, 19 

Charles's Law, 47 

Chart recorder, 142 

Chemical seal, 331 

Chemical versus nuclear reaction, 64 

Chromatogram, 564 

Chromatography, 561 

CIP, 330 

Cippoletti weir, 489, 582 

Cistern manometer, 291 

Clamp-on milliammeter, 203 

Class I filled system, 419 

Class II filled system, 420 

Class III filled system, 419 

Class V filled system, 419 

Clean-In-Place, 330 

Cold junction compensation, 428 

Cold junction, thermocouple, 429 

Column, chromatograph, 561 

Combination electrode, 553 

Combustion, 65 

Common-mode rejection, 320 

Compensating leg, 367 

Complex number, 121 

Compound, 61 

Compressibility, 48 

Compression fitting, 318 

Condensate boot, 382 

Conductance, 92, 99 

Conductivity cell, 543 
Conductivity sensor, 543 
Conical-entrance orifice plate, 463 
Conservation of Electric Charge, 90 
Conservation of Energy, 20, 23, 30, 55, 91, 93, 

Conservation of Mass, 20, 53, 64, 96, 449 
Constant of proportionality, 453 
Control algorithm, 601 
Controller, 131 
Controller gain, 604 
Conventional flow, 175 
Coriolis force, 515 
Coriolis mass flowmeter, 516 
Corner taps (orifice plate), 465 
Coulomb, 75, 80 

Counterpropagation ultrasonic flowmeter, 512 
cps, 275 

Crank diagram, 119 
Crest, weir, 490 
Current, 79, 80 
Curved manometer, 578 
Custody transfer, 375, 477, 499, 523, 527 
Cycles per second, 275 

Dall flow tube, 469 

DC excitation, magnetic flowmeter, 511 

Dead-test unit, 278 

Deadweight tester, 278 

Deadweight tester, pneumatic, 280 

Degrees API, 38 

Degrees Balling, 39 

Degrees Bark, 39 

Degrees Baume, 38 

Degrees Brix, 39 

Degrees Oleo, 39 

Degrees Soxhlet, 39 

Degrees Twaddell, 38 

Density, influence on hydrostatic level 

measurement accuracy, 358 
Dependent current source, 196 
Derivative control, 614 
Derivative notation, calculus, 530 
Derived unit, 19 
Diaphragm, 223, 295 
Diaphragm, isolating, 301, 303, 306, 330 



Dielectric constant, 397 

Dielectric constant, influence on radar level 

measurement accuracy, 401 
Differential, 614 

Differential capacitance pressure sensor, 303 
Differential notation, calculus, 530 
Differential pressure, 43, 294 
Differential temperature sensing circuit, 106 
Differentiation, applied to capacitive voltage 

and current, 123 
Digital multimeter, 275 
Dimensional analysis, 18, 23, 37 
Diode, in current loop circuit, 203 
Dip tube, 361 

Direct-acting controller, 604 
Direct-acting pneumatic relay, 225 
Direct-acting transmitter, 154 
Discharge coefficient, 477 
Discrete, 169, 266 
Displacement, 45 
Displacer level instrument, 382 
Dissociation, 542 
DMM, 275 
Doppler effect, 512 
Doppler ultrasonic flowmeter, 512 
DP cell, 239 
Drift, 270 
Droop, 610 
Dry leg, 368 
Dry-block temperature calibrator, 276 

Eccentric orifice plate, 460 

Einstein, Albert, 20 

Electrical heat tracing, 341 

Electrodeless conductivity cell, 546 

Electrolysis, 65 

Electromagnetic induction, 505 

Electron flow, 175 

Electron shell configuration, 63 

Electronic manometer, 283 

Element, 61 

Emerson AMS software, 254, 590 

Emerson DeltaV control system, 254, 590 

Emissivity, thermal, 435 

Emittance, thermal, 435 

Endothermic, 65 

Endress+Hauser magnetic flowmeter, 509 

Equivalent circuits, series and parallel AC, 117 

Error, controller, 604, 611 

Euler's relation, 122 

Excitation source, for bridge circuit, 100 

Exothermic, 65 

Farad, 108 

Feedback control system, 600 

Fieldbus, 139, 254 

Fill fluid, 301, 303, 306, 325, 328, 419 

Fillage, 390 

Filled bulb, 32, 419 

Filled impulse line, 337 

Filtering, 617 

Final Control Element, 131 

First Law of Motion, 21 

Fisher "LevelTrol" displacer instrument, 382 

Five-point calibration, 265 

Five-valve manifold, 323 

Flame ionization detector, GC, 563 

Flange taps (orifice plate), 465 

Flapper, 213 

Flexure, 269 

Float level measurement, 352 

Flow conditioner, 474 

Flow prover, 284 

Flow switch, 185 

Flow tube, 469 

Flow-straightening vanes, 474 

Fluid, 27, 28 

Flume, 491, 581 

Force balance system, 234, 312 

Form- A contact, 170 

Form-B contact, 170 

Form-C contact, 173 

Foxboro magnetic flowtube, 510 

Foxboro model 

13 differential pressure transmitter, 

Foxboro model 557 pneumatic square root 

extractor, 457 
Foxboro model IDP10 differential pressure 

transmitter, 302, 316 
Freezing point of water, 275 
Frequency shift keying, 246 



FSK, 246 

Full-active bridge circuit, 106 
Full- flow taps (orifice plates), 466 
Function, inverse, 573 
Function, piecewise, 587 

Gain, controller, 604, 616 

Galilei, Galileo, 21 

Gas, 28 

Gas expansion factor, 477 

Gas Laws, 47 

Gas, calibration, 287 

Gas, span, 287 

Gauge line, 318 

Gauge pressure, 43, 294 

Gauge tube, 318 

Gay-Lussac's Law, 47 

Generator, 80 

Gentile flow tube, 469 

Gerlach scale, 38 

Ground, 96 

Grounding, magnetic flowmeters, 509 

Guided wave radar, 395 

Hagen-Poiseuille equation, 54, 485, 628 

Hall Effect sensor, 311 

Hand switch, 172 

HART multidrop mode, 252 

Head (fluid), 55 

Heat exchanger, 596 

Heat tape, 341 

Heat tracing, 340 

Helical bourdon tube, 283, 295 

Henry, 110 

Herschel, Clemens, 469 

Hertz, 275 

Hot-tapping, 533 

Hot-wire anemometer, 524 

HVAC, 424 

Hydration, pH electrode, 555 

Hydraulic, 31 

Hydraulic lift, 29 

Hydraulic load cell, 405 

Hydrogen economy, 65 

Hydrogen ion, 68, 69, 542 

Hydronium ion, 68, 69, 542 

Hydrostatic pressure, 36 
Hydroxyl ion, 68, 69, 542 
Hysteresis, 265 

I/P transducer, 204, 212 

Ice point, thermocouple, 432 

Ideal Gas Law, 47 

Ideal PID algorithm, 617 

Impedance, 117, 127 

Impulse line, 318 

Impulse tube, 318, 325 

Inches of mercury, 36 

Inches of water column, 36 

Inclined manometer, 40, 291 

Indicator, 141 

Inductance, 110 

Inductor, 110 

Inferential measurement, 284, 289, 374, 574 

Inferred variable, 374, 574 

Instrument tube bundle, 340 

Integral control, 611 

Integral orifice plate, 466 

Integral windup, 613 

Integration, applied to RMS waveform value, 

Interactive zero and span adjustments, 260, 

Interface level measurement, 347 
Intrinsic safety, 315 
Intrinsic standard, 271 
Inverse function, 573 
Inviscid flow, 51 
Ion, 61 

Ion-selective membrane, 592 
Ionization, 542 
ISA PID algorithm, 617 
Isolating diaphragm, 301, 303, 306, 330 
Isopotential point, pH, 560 
Isotope, 62 

Joule, 75 
Joule's Law, 99 

KCL, 96 

Kelvin resistance measurement, 424, 545 

Kinematic viscosity, 49 



Kinetic energy, 22 
Kirchhoff's Current Law, 96 
Kirchhoff 's Voltage Law, 94 
Knockout drum, 382 
KVL, 94 

Laminar flow, 52, 54, 628 

Laminar flowmeter, 485 

Law of Continuity (fluids), 53, 449, 495 

Law of Intermediate Metals, thermocouple 

circuits, 430 
Level gauge, 348 
Level switch, 181 
Limit switch, 173 
Linearity error, 268 
Linearization, 580 
Liquid, 28 

Liquid interface detection with radar, 399 
Live zero, 259 
Lo-Loss flow tube, 469 
Load, 80, 608 
Load cell, 104, 403 
Load cell, hydraulic, 405 
Load versus source, 98 
Loop-powered transmitter, 200 
Lower range value, 131, 259, 262, 283 
LRV, 131, 259, 262, 283 
LVDT, 311 

Magnetic flowmeter, 506 
Magnetrol liquid level switch, 181 
Manifold, pressure transmitter, 322, 323 
Manipulated variable, 598 
Manometer, 40, 281, 290 
Manometer, cistern, 291 
Manometer, inclined, 40, 291 
Manometer, nonlinear, 578 
Manometer, raised well, 291 
Manometer, slack tube, 283 
Manometer, U-tube, 291 
Manometer, well, 291 
Manual mode, 131 
Mass density, 9 
Mass flow, 443 

Maximum working pressure, 320 
Measurement electrode, 550 

Measurement junction, thermocouple, 428 

MEMS, 308 

Meniscus, 290 

Mercury, 345 

Mercury barometer, 294 

Metal fatigue, 300 

Metrology, 271 

Micromanometer, 41 

Minutes per repeat, 616 

Mixture, 61 

Mobile phase, 561 

Molarity, 63, 285 

Mole, 63 

Molecule, 61 

Moment balance system, 234 

Motion balance system, 234 

Motional EMF, 505 

Multi-segment characterizer, 587 

Multi- variable transmitter, 253, 399, 481, 522, 

Multidrop, HART, 252 
Multipath ultrasonic flowmeter, 513 
MWP, 320 

NBS, 271 

Needle valve, 327 

Nernst equation, 550, 556, 592 

Newton, Isaac, 21 

NIST, 271 

Non-bleeding pneumatic relay, 227 

Non-contact radar, 395 

Non-inertial reference frame, 515 

Non-Newtonian fluid, 50 

Nonlinear manometer, 578 

NOx emissions, 592 

Nozzle, 213 

Nuclear versus chemical reaction, 64 

Ohm, 87 

Ohm's Law, 99 

Ohm, Georg Simon, 87 

Oil bath temperature calibrator, 275 

Oleo, degrees, 39 

On-off control, 602 

Order of magnitude, 230 

Orifice plate, 458, 574 



Orifice plate, concentric, 459 
Orifice plate, conical entrance, 463 
Orifice plate, eccentric, 460 
Orifice plate, integral, 466 
Orifice plate, quadrant edge, 463 
Orifice plate, segmental, 461 
Orifice plate, square-edged, 459 
Oxygen control, burner, 592 

Parallel PID algorithm, 616 

Parshall flume, 581 

Particle, 61 

Parts per million, 287 

Pascal, 29 

Pascal's principle, 33 

Periodic table of the elements, 62 

Permanent pressure loss, 58 

Permittivity, 108 

Permittivity, relative, 397 

pH, 69, 285 

Phase change, 275 

Phasor, 122 

Pickup coil, 497 

Piecewise function, 587 

Piezometer, 449 

Pigtail siphon, 342 

Pilot valve, 221 

Pipe elbow flow element, 472 

Pipe hanger, 404 

Pipe taps (orifice plate), 466 

Pitot tube, 467 

Pneumatic, 31 

Pneumatic "resistor" , 486 

Pneumatic control system, 135 

Pneumatic deadweight tester, 280 

Pneumatic relay, 224 

Poise, 49 

Polarity, 77 

Position PID algorithm, 617 

Potential energy, 22, 74 

Power reflection factor, 398 

Powers and roots, 582 

ppm, 287 

Preamplifier, pH probe, 558 

Predictive maintenance, 270 

Pressure, 27, 29, 344 

Pressure gauge mechanism, typical, 295 
Pressure recovery, 58 
Pressure snubber, 327 
Pressure switch, 179 
Pressure, absolute, 43 
Pressure, differential, 43 
Pressure, gauge, 43 
Pressure, hydrostatic, 36 
Pressure-based flowmeters, 444 
Primary sensing element, 131 
Process, 130, 596 
Process switch, 144 
Process variable, 130, 597 
Programming, chromatograph, 571 
Projectile physics, 23 
Proportional band, controller, 616 
Proportional control, 603 
Proportional weir, 491 
Proportional-only offset, 610 
Proximity switch, 175 
Purge flow rate, 339, 361 
Purged impulse line, 339 

Quadrant-edge orifice plate, 463 
Quarter-active bridge circuit, 106 

Radar detection of liquid interfaces, 399 

Radar level instrument, 395 

Radioactivity, 62 

Raised well manometer, 291 

Range wheel, 239 

Rangeability, 557 

Rangedown, 271 

Ranging, 257 

Rate control, 614 

Reactance, 117 

Real Gas Law, 48 

Receiver gauge, 214 

Recorder, 142 

Rectangular weir, 489 

Reference electrode, 552 

Reference junction compensation, 428 

Reference junction, thermocouple, 428 

Reflection factor, 398 

Relative permittivity, 397 



Relative permittivity, influence on radar level 

measurement accuracy, 401 
Remote seal, 331 
Repeats per minute, 016 
Repose, angle of, 392 
Reset control, 611 
Reset windup, 613 
Resistance, 87, 99, 117 
Resistor, 99 

Resonant wire pressure sensor, 308 
Reverse- acting controller, 604 
Reverse-acting pneumatic relay, 225 
Reverse-acting transmitter, 154 
Reynolds number, 51 
Richter scale, 557 
Roots and powers, 582 
Rosemount Micro-Motion Coriolis mass 

flowmeter, 519 
Rosemount model 1151 differential pressure 

transmitter, 304, 316, 358 
Rosemount model 3051 differential pressure 

transmitter, 207, 306, 316, 363, 481 
Rosemount model 3095MV multi-variable 

transmitter, 482 
Rosemount model 3301 guided- wave radar 

transmitter, 590 
Rosemount model 8700 magnetic flowmeter, 

Rotameter, 361, 487 
RTD, 275, 422 

Salt, 72 

SAMA diagram, 156 

Sand bath temperature calibrator, 275 

Second Law of Motion, 21, 45 

Segmental orifice plate, 461 

Segmental wedge, 471 

Self-balancing bridge, 102 

Self-balancing system, 218, 312 

Self-powered transmitter, 198 

Sensing line, 318 

Sensing tube, 318 

Series PID algorithm, 617 

Setpoint, 130, 599 

Setpoint tracking, 158 

Shelf life, pH electrode, 555 

Sightfeed bubbler, 361 

Sightglass, 348 

Silicon resonator pressure sensor, 308 

Sinking output switch, 175 

SIP, 330 

Slack diaphragm, 295 

Slack-tube manometer, 283 

Slope, pH instrument, 559 

Smart instrument, 261 

Smart transmitter, 207 

Snubber, pressure, 327 

Solid, 28 

Sonic level instrument, 390 

Source versus load, 98 

Sourcing output switch, 175 

Soxhlet, degrees, 39 

Span adjustment, 259 

Span gas, 287 

Span shift, 267 

Specific gravity, 38, 46 

Specific heat, 526 

Spiral bourdon tube, 295 

Square root characterizer, 456, 579, 623, 625, 

Square root extractor, 457 
Square root scale, 576 
Square-edged concentric orifice plate, 459 
Stagnation pressure, 447 
Standard cell, 273 
Stationary phase, 561 
Steam jacket, 139 
Steam tracing, 340 
Steam trap, 340 
Steam-In-Place, 330 
Stefan-Boltzmann equation, 591 
Stefan-Boltzmann Law, 435 
Steinmetz, Charles Proteus, 122 
Stem valve, 226 
Stilling well, 409, 493 
Stoichiometry, 64 
Stokes, 50 

Strain gauge, 103, 300 
Strapping table, 589 
Strouhal number, 501 
Strouhal, Vincenc, 501 
Superconductivity, 87 



Superfluidity, 87 
Sutro weir, 491 
Swamping, 423, 628 
Switch, 169 
Switch, process, 144 
Systeme International, 19 

Tank expert system, 372 

Tape-and-float level measurement, 354 

Tare weight, 403 

Target flow element, 469 

Temperature switch, 183 

Temperature, defined for a gas, 415 

Test Uncertainty Ratio, 272 

Thermal conductivity detector, GC, 564 

Thermal energy, 415 

Thermal imager, 435 

Thermal mass flowmeter, 524 

Thermistor, 422 

Thermocouple, 275, 427 

Thermocouple burnout detection, 433 

Thermowell, 437 

Thin- layer chromatography, 561 

Third Law of Motion, 21 

Three-valve manifold, 322 

Toroidal conductivity cell, 546 

Torr, 44 

Torricelli, Evangelista, 57 

Toshiba magnetic flowmeter, 509 

Transducer, 131 

Transit-time ultrasonic flowmeter, 512 

Transmitter, 131 

Trap, 342 

Trap, steam, 340 

Trend recorder, 142 

Tuning, controller, 616 

TUR, 272 

Turbine flow element, 496 

Turbulent flow, 52 

Turndown, 271 

Twaddell, degrees, 38 

U-tube manometer, 291 
Ullage, 390 

Ultrasonic flowmeter, 512 
Ultrasonic level instrument, 390 

Unit conversions, 10 

Unity fraction, 10 

Up-down calibration test, 265 

Upper range value, 131, 262, 283 

URV, 131, 262, 283 

V-cone flow element, 470 

V-notch weir, 489 

Variable-area flowmeter, 487 

Velocity of approach factor, 483 

Velocity PID algorithm, 617 

Vena contracta, 458, 622 

Venturi tube, 58, 449 

Viscosity, 49 

Viscosity, absolute, 49 

Viscosity, kinematic, 49 

Viscosity, temperature dependence, 50 

Viscous flow, 51 

Volt, 75 

Volta, Alessandro, 75 

Voltage, 74 

Volumetric flow, 443 

von Karman, Theodore, 501